# Calculus 2 : Series and Functions

## Example Questions

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### Example Question #11 : Introduction To Series In Calculus

What do we mean when we say an infinite series converges?

The sequence partial sums of the sequence , denoted converges as .

The sequence partial sums of the sequence converges as .

The sequence partial sums of the sequence , also denoted converges as .

None of the other choices

The sequence converges as .

The sequence partial sums of the sequence , denoted converges as .

Explanation:

This is the definition of a convergent infinite series.

### Example Question #12 : Introduction To Series In Calculus

What is the sum of the following geometric series:     Cannot be determined. Explanation:

Since this is a geometric series with a rate between and , we can use the following equation to find the sum: , where is the starting number in the sequence, and is the common divisor between successive terms in the sequence.  In this sequence, to go from one number to the next, we multiply by Now, we plug everything into the equation: ### Example Question #13 : Introduction To Series In Calculus

Find the infinite sum of the following geometric series:    Cannot be determined  Explanation:

Since this is a geometric series with a rate between and , we can use the following equation to find the sum: , where is the starting number in the sequence, and is the common divisor between successive terms in the sequence.  In this sequence, to go from one number to the next, we multiply by Now, we plug everything into the equation: ### Example Question #14 : Introduction To Series In Calculus

Find the infinite sum of the following series:       Explanation:

For the sum of an infinite series, we have the following formula: , where is the first term in the series and is the rate at which our series is changing between consecutive numbers in the series.  Plugging all of the relevant information for this series, we get: ### Example Question #15 : Introduction To Series In Calculus

Find the infinite sum of the following series:       Explanation:

For the sum of an infinite series, we have the following formula: , where is the first term in the series and is the rate at which our series is changing between consecutive numbers in the series.  Plugging all of the relevant information for this series, we get: ### Example Question #16 : Introduction To Series In Calculus

A) Find a power series representation of the function, B) Determine the power series radius of convergence.

A) Power Series for , 2

A) Power Series for  A) Power Series for  A) Power Series for , 1

A) Power Series for , 1

A) Power Series for  Explanation: This function can be easily written as a power series using the formula for a convergent geometric series.

____________________________________________________________ For any ____________________________________________________________

First let's make some modifications to the function so we can compare it to the form of a convergent geometric series: Notice if we take and we can write in the form, We can find the radius of convergence by applying the condition .

_____________________________________________________________ Case 1 Case 2 Combing both cases gives the interval of convergence, Therefore the radius of convergence is ____________________________________________________________

We can continue simplifying our most recent expression of   ### Example Question #17 : Introduction To Series In Calculus

Does the following series converge or diverge: Diverge

Converge

Cannot be determined with the given information.

Diverge

Explanation:

To test if this series diverges, before using a higher test, we may use the test for divergence.

The test for divergence informs that if the sequence does not approach 0 as n approaches infinity then the series diverges (NOTE: This only shows divergence, the converse is not true, that is, the test for divergence cannot be used to show convergence.).

We note that as ,

this is derived from the fact that to find the limit as x approaches infinity of a function, one must first find the horizontal asymptote. Since this function is a rational expression with the highest power in both the numerator and denominator, the horizontal asymptote is equal to the quotient of the leading coefficients of both the numerator and denominator, which in this case is 2/3.

Since the limit as x tends to infinity of this series is a nonzero value, we may conclude that the series diverges by the Test for Divergence.

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