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# Category Archives: Riemann Sum

## Total Number of Triangles Puzzle

How many triangles are there in the figure below?

Solution to this Geometry Triangle Puzzle problem is given in the video below!

## Total Number of Rectangles Puzzle

How many rectangles are there in the figure below?

Solution to this Geometry Rectangle Puzzle is given in the video below!

## Total Number of Upright SQUARES in a Square

How many upright Squares are there in the following square?

The key is to understand vertices of any given upright square and a diagonal line of the largest square to see how many translations of that given upright square you can have. As such, there is a summation-derived FORMULA you can actually use for any largest size square to find the total number of possible upright squares contained within it!

Solution to this Geometry Square Puzzle is given in the video below!

## Total Number of Upright RECTANGLES in a Square (or Rectangle)

How many upright Rectangles are there in the following square?

The key is to understand vertices of any given upright rectangle to see how many unique translations of that rectangle are possible. As such, there is actually a summation-derived FORMULA you can utilize for the largest square or rectangle of any size to find the total number of possible upright rectangles within it!

Solution to this Geometry Rectangle Puzzle is given in the video below!

## Total Number of Upright SQUARES in a Rectangle

How many upright Squares are there in the following rectangle?

The key is to understand vertices of any given upright square to see how many translations of that given square you can have. As such, there is a summation-related FORMULA you can actually derive for any size rectangle to find the total number of possible upright squares contained within it!

Solution to this Square Rectangle Geometry Puzzle is given in the video below!

## Total Number of Upright & Tilted SQUARES in a Square

How many upright and tilted Squares are there in the following square?

The key is to use vertices of any given largest square to see how many translations of a given smaller square you can have. As such, there is a summation-derived FORMULA you can actually use for any size square to find the total number of possible upright and tilted squares contained within it!

Solution to this Square Geometry Puzzle example is given in the video below!

## Total Number of Upright & Tilted SQUARES in a Rectangle

How many upright and tilted Squares are there in the following rectangle?

The key is to use vertices of any given rectangle to see how many translations of a given square you can have. As such, there is a summation-derived FORMULA you can actually use for any size rectangle to find the total number of possible upright and tilted squares contained within it!

Solution to this Rectangle Geometry Puzzle example problem is given in the video below!

## Nth Term of a Sequence example problem

Find a formula for the Nth Term of a Sequence whose first four terms are given by

(a)  4, 7, 10, 13, …

(b)  1, -3, 5, -7, …

(c)

(d)

Solution to these Arithmetic and Geometric Sequences practice problems is provided in the video below!

## Sequence Terms and Summation Notation example question

Write the following Sum of the Terms of a Sequence in Summation Notation:

(a)

(b)

(c)

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!

## Testing For Arithmetic and Geometric Sequences example

Determine if the following Sequences are Arithmetic, geometric or Neither:

(a)

(b)

(c)

(d)

(e)

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!

## Arithmetic Sequence Common Difference Nth term example problem

For the following Arithmetic sequences, state the common difference, and write the next three terms and the Nth term:

(a)

(b)  -8, -5, …

(c)

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!

## Geometric Sequence Common Ratio Nth term example question

For the following Geometric sequences, state the common ratio, and write the next three terms and the Nth term:

(a)

(b)  -5, 5, -5, …

(c)  1, 1.05, …

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!

## Infinite Geometric Sequence SUM example

For the following Geometric sequences, state the common ratio and find the Sum of ALL Terms (Infinite Sum), or state that the sum is undefined:

(a)  4,

(b)

(c)  36, -12, 4, …

(d)  1, 0.95, …

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!

## Geometric Sequence application word problem

Suppose that \$0.01 were deposited into a bank account on the first day of June, \$0.02 on the second day, \$0.04 on the third day, and so on in a geometric sequence.

(a) How much money would be deposited at this rate on June 30th?

(b) How much money would be in the account after this last deposit?

Solution to this Arithmetic and Geometric Sequences problem is provided in the video below!

## Infinite Series Proof example question

Show that

when

Solution to this Calculus & Precalculus Infinite Series Proof practice problem is given in the video below!

## Fourier Series example question

Find the Fourier series representation of the function in the given interval.

Solution to this Calculus & Precalculus Fourier Infinite Series practice problem is given in the video below!

## Fourier Series example problem

Find the Fourier series representation of the given function and its interval.

Solution to this Calculus & Precalculus Fourier Infinite Series practice problem is given in the video below!

## Fourier Series example

Find the Fourier series of the function in the given interval.

Solution to this Calculus & Precalculus Fourier Infinite Series practice problem is given in the video below!

## Fourier Series problem example

Find the Fourier series representation of the function on the given interval below.

Solution to this Calculus & Precalculus Fourier Infinite Series practice problem is given in the video below!

## Fourier Series example question #5

Find the Fourier series representation of the function with the given interval.

Solution to this Calculus & Precalculus Fourier Infinite Series practice problem is given in the video below!

## Fourier Series example #6

Determine the Fourier series of the function in the given interval.

Solution to this Calculus & Precalculus Fourier Infinite Series practice problem is given in the video below!

## Fourier Series example problem #7

Determine the Fourier series representation of the function in the given interval.

Solution to this Calculus & Precalculus Fourier Infinite Series practice problem is given in the video below!

## Maclaurin Series Interval of Convergence example question

Find the Maclaurin series (Taylor series about c = 0) and its interval of convergence for the function

Solution to this Calculus & Precalculus Maclaurin Infinite Series practice problem is given in the video below!

## Maclaurin Series Interval of Convergence example problem

Determine the Maclaurin series (Taylor series about c = 0) and its interval of convergence for the function

Solution to this Calculus & Precalculus Maclaurin Infinite Series practice problem is given in the video below!

## Maclaurin Series Interval of Convergence example #3

Find the Maclaurin series (Taylor series about c = 0) and its interval of convergence for the given function

Solution to this Calculus & Precalculus Maclaurin Infinite Series practice problem is given in the video below!

## Maclaurin Series Interval of Convergence example question #4

Determine the Maclaurin series (Taylor series about c = 0) and its interval of convergence for the given function

Solution to this Calculus & Precalculus Maclaurin Infinite Series practice problem is given in the video below!

## Taylor Series Interval of Convergence example problem #1

Determine the Taylor series about c = 1 and its interval of convergence for the given function

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Interval of Convergence example #2

Determine the Taylor series about c = 1 and its interval of convergence for the function

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Interval of Convergence example question #3

Determine the Taylor series about c e and its interval of convergence for the following function

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Radius of Convergence example problem #1

Determine the Taylor series about c = 0 and its radius of convergence for the given function

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Radius of Convergence example #2

Find the Taylor series about c = 0 and its radius of convergence for the given function

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Radius of Convergence example question #3

Determine the Taylor series about c = 0 and its radius of convergence for the following function

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Limit example problem #1

Use the Taylor series expansion about c = 0 to find the indicated Limit.

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Limit example #2

Use the Taylor series expansion about c = 1 to find the given Limit.

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Limit example question #3

Use the Taylor series expansion about c = 0 to evaluate the given Limit.

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Definite Integral example problem #1

Use the Taylor series expansion about c = 0 to evaluate the Definite Integral.

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Definite Integral example #2

Use the Taylor series expansion about c = 0 to find the Definite Integral value.

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Taylor Series Definite Integral example question #3

Use the Taylor series expansion about c = 1 to evaluate the following Definite Integral.

Solution to this Calculus & Precalculus Taylor Infinite Series practice problem is given in the video below!

## Non-Power Series Interval of Convergence & Derivatives example question

Find the interval of convergence of the non-power series

and the corresponding series of derivatives.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Non-Power Series Interval of Convergence & Derivatives example problem

Find the interval of convergence of the following non-power series

and determine the corresponding series of derivatives.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Representation of Functions example question

Find a power series representation of

about c = 0 and determine the radius and interval of convergence.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Representation of Functions example problem

Find a power series representation of

about c = 0 and determine the radius and interval of convergence.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Representation of Functions example

Find a power series representation of

about c = 0 and determine the radius and interval of convergence.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Representation of Functions example question #4

Find a power series representation of

about c = 0 and determine the radius and interval of convergence.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Representation of Functions example problem #5

Find a power series representation of

about c = 0 and determine the radius of convergence by integrating the derived function.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Representation of Functions example #6

Find a power series representation of

about c = 0 and determine the radius of convergence by taking derivative of the given function.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Representation of Functions example question #7

Find a power series representation of

about c = 0 and determine the radius of convergence by integrating the given function.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Function Interval of Convergence example problem

Determine the interval of convergence of the power series

and the function to which it converges.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Function Interval of Convergence example #2

Find the interval of convergence of the power series

and the function to which it converges.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Function Interval of Convergence example question #3

Determine the interval of convergence of the power series

and the function of convergence.

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Interval of Convergence example problem #1

Determine the interval and radius of convergence of the power series

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Interval of Convergence example #2

Find the interval and radius of convergence of the power series

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Interval of Convergence example problem #3

Determine the interval and radius of convergence of the given power series

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Interval of Convergence example question #4

Find the interval and radius of convergence of the given power series

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Interval of Convergence example #5

Determine the interval and radius of convergence of the power series given below

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Interval of Convergence example problem #6

Find the interval and radius of convergence of the power series given below

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Interval of Convergence example question #7

Determine the radius and interval of convergence of the power series

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Power Series Radius Interval of Convergence example #8

Find the radius and interval of convergence of the power series

Solution to this Calculus & Precalculus Power Infinite Series practice problem is given in the video below!

## Absolute Convergence Test Series example question

Determine whether the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example problem

Determine if the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example

Determine whether the infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example question #4

Determine if the infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example problem #5

Determine whether the following infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example #6

Determine if the following infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example question #7

See if the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example problem #8

See whether the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example #9

Determine whether the infinite series given below

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example question #10

Determine whether the provided infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example problem #11

Determine whether the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example #12

Does the infinite series

converge absolutely, converge conditionally, or diverge?

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example question #13

Determine whether the infinite series given below

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example problem #14

Determine if the given infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Absolute Convergence Test Series example #15

Determine if the following infinite series

is absolutely convergent, conditionally convergent, or divergent.

Solution to this Calculus & Precalculus Absolute Convergence Test Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example problem

Determine if the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example

Determine whether the infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example question #4

Determine if the given infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example problem #5

See whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example #6

See if the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example question #7

See whether the infinite series

converges or diverges.

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example problem #8

Does the given infinite series

converge or diverge?

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example #9

Does the infinite series

converge or diverge?

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example question #10

Is the given infinite series

convergent or divergent?

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example problem #11

Is the infinite series

convergent or divergent?

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example #12

Determine if the infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Alternating Series Convergence Divergence example question #13

Determine if the following infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Alternating Series practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Limit Comparison Test Series practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example problem

Determine if the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Limit Comparison Test Series practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example

Determine whether the given infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Limit Comparison Test Series practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example question #4

Determine if the infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Limit Comparison Test Series practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example problem #5

Does the given infinite series

converge or diverge?

Solution to this Calculus & Precalculus Limit Comparison Test Series practice problem is given in the video below!

## Limit Comparison Test Series Convergence Divergence example #6

Is the given infinite series

convergent or divergent?

Solution to this Calculus & Precalculus Limit Comparison Test Series practice problem is given in the video below!

## Comparison Test Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Comparison Test Series practice problem is given in the video below!

## Comparison Test Series Convergence Divergence example problem

Does the given infinite series

converge or diverge?

Solution to this Calculus & Precalculus Comparison Test Series practice problem is given in the video below!

## Comparison Test Series Convergence Divergence example

Determine whether the given infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Comparison Test Series practice problem is given in the video below!

## Integral Test Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Integral Test Series practice problem is given in the video below!

## Integral Test Series Convergence Divergence example problem

Determine if the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Integral Test Series practice problem is given in the video below!

## Integral Test Series Convergence Divergence example

Determine whether the given infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Integral Test Series practice problem is given in the video below!

## Integral Test Series Convergence Divergence example question #4

Determine if the given infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Integral Test Series practice problem is given in the video below!

## Integral Test Series Convergence Divergence example problem #5

Does the given infinite series

converge or diverge?

Solution to this Calculus & Precalculus Integral Test Series practice problem is given in the video below!

## Harmonic Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Harmonic Series practice problem is given in the video below!

## P Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus P Series practice problem is given in the video below!

## P Series Convergence Divergence example problem

Determine if the given infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus P Series practice problem is given in the video below!

## Telescoping Infinite Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Telescoping Infinite Series practice problem is given in the video below!

## Telescoping Infinite Series Convergence Divergence example problem

Determine whether the given infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Telescoping Infinite Series practice problem is given in the video below!

## Telescoping Infinite Series Convergence Divergence example

Is the infinite series

convergent or divergent?

Solution to this Calculus & Precalculus Telescoping Infinite Series practice problem is given in the video below!

## Multiple terms Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Infinite Series practice problem is given in the video below!

## Multiple terms Convergence Divergence example problem

Does the given infinite series

converge or diverge?

Solution to this Calculus & Precalculus Infinite Series practice problem is given in the video below!

## Nth Term Test Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Nth Term Test Infinite Series practice problem is given in the video below!

## Geometric Series Convergence Divergence example question

Determine whether the given infinite series

converges or diverges.

Solution to this Calculus & Precalculus Geometric Infinite Series practice problem is given in the video below!

## Geometric Series Convergence Divergence example problem

Determine whether the infinite series

converges or diverges.

Solution to this Calculus & Precalculus Geometric Infinite Series practice problem is given in the video below!

## Geometric Series Convergence Divergence example

Determine whether the given infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Geometric Infinite Series practice problem is given in the video below!

## Geometric Series Convergence Divergence example question #4

Determine if the infinite series

is convergent or divergent.

Solution to this Calculus & Precalculus Geometric Infinite Series practice problem is given in the video below!

## Riemann Sum Integration example question

Find the Definite Integral of a function f(x) = x2 – 3 from x = 1 to x = 3, by using the Riemann Sum.

Solution to this Calculus Riemann Sum Definite Integral practice problem is given in the video below!

## Sequences & Series problems (complete Playlist)

Lots of different Sequences & Series examples and practice problems applicable to your Calculus or Precalculus courses.