Calculus 2 : Series and Functions

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #11 : Series And Functions

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

Possible Answers:

The sequence  converges as .

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

None of the other choices

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

The sequence partial sums of the sequence   converges as .

Correct answer:

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

Explanation:

This is the definition of a convergent infinite series.

Example Question #12 : Series And Functions

What is the sum of the following geometric series:

 

Possible Answers:

Cannot be determined.  

Correct answer:

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 : Series And Functions

Find the infinite sum of the following geometric series:

Possible Answers:

Cannot be determined

Correct answer:

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 : Series And Functions

Find the infinite sum of the following series:

Possible Answers:

Correct answer:

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 : Series And Functions

Find the infinite sum of the following series:

Possible Answers:

Correct answer:

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 : Series And Functions

A) Find a power series representation of the function, 

  

 

B) Determine the power series radius of convergence.  

Possible Answers:

A) Power Series for ,

 

 

B) Radius of Convergence 

 2 

 

A) Power Series for 

 

 

 

B) Radius of Convergence 

 

A) Power Series for 

 

 

B) Radius of Convergence 

 

A) Power Series for ,

 

 

 

 

B) Radius of Convergence 

 1

 

A) Power Series for ,

 

 

B) Radius of Convergence 

 1

 

Correct answer:

A) Power Series for 

 

 

B) Radius of Convergence 

 

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  .

 _____________________________________________________________

Finding Radius of Convergence. 

 

 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 : Series And Functions

Does the following series converge or diverge:

Possible Answers:

Converge

Cannot be determined with the given information.

Diverge

Correct answer:

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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