Calculus 2 : Arithmetic and Geometric Series

Study concepts, example questions & explanations for Calculus 2

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Example Questions

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Example Question #1 : Arithmetic And Geometric Series

How many terms of a geometric series must you know in order to uniquely define that series?

Possible Answers:

Four

Two

Five

One

Three

Correct answer:

Two

Explanation:

In order to uniquely define the geometric series, we need to know two things: the ratio between successive terms and at least one of the terms. Knowing one term doesn't give you the ratio of successive terms, but knowing two terms will give you the ratio. By the term generator for a geometric series , you can see that you only need two terms to find the ratio 

Example Question #1 : Arithmetic And Geometric Series

Assume the term generator for an arithmetic sequence is . What is the sum of the first  terms of this sequence ?

Possible Answers:

Correct answer:

Explanation:

The sum formula for  terms of an arithmetic series is .

For  terms, this formula becomes .

Using our term generator for  and , this formula becomes 

.

Example Question #2 : Arithmetic And Geometric Series

What value does the series  approach?

Possible Answers:

Correct answer:

Explanation:

We can evaluate the infinite series by recognizing it as a geometric series times some constant.

Let's manipulate this series:

.

Now it suffices to evaluate , which we can recognize as the power series of  with , which is

 .

So we have

.

Example Question #4 : Arithmetic And Geometric Series

Determine whether the series is arithmetic. If so, find the common difference.

Possible Answers:

Series is not arithmetic

Correct answer:

Series is not arithmetic

Explanation:

If a series is arithmetic, then there exists a common difference between each pair of consecutive terms in the series.

For this series

Because 

and

we find that there does NOT exist a common difference and as such,

the series is not arithmetic.

Example Question #5 : Arithmetic And Geometric Series

Determine whether or not the geometric series converges. If it converges, find the sum of the sequence.

Possible Answers:

Series does not converge.

Correct answer:

Series does not converge.

Explanation:

To determine the convergency of a geometric series, we must find the absolute value of the common ratio.

A geometric series will converge if the absolute value of the common ratio is less than one, or

In this problem, we see that 

And because 

we conclude that the series does not converge to a finite sum.

Example Question #6 : Arithmetic And Geometric Series

Calculate the sum of the following series: 

Possible Answers:

Correct answer:

Explanation:

This is an arithmetic series.

Its general form is .

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

Solution:

Example Question #3 : Arithmetic And Geometric Series

Calculate the sum of the following series: 

Possible Answers:

Correct answer:

Explanation:

This is an arithmetic series.

Its general form is 

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

Solution:

Example Question #4 : Arithmetic And Geometric Series

Calculate the sum of the following series: 

Possible Answers:

Correct answer:

Explanation:

This is an arithmetic series.

Its general form is 

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

Solution:

Example Question #5 : Arithmetic And Geometric Series

Calculate the sum of the following series: 

Possible Answers:

Correct answer:

Explanation:

This is an arithmetic series.

Its general form is 

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

Solution:

Example Question #6 : Arithmetic And Geometric Series

Calculate the sum of the following series: 

Possible Answers:

Correct answer:

Explanation:

This is an arithmetic series.

Its general form is 

To calculate the sum of very large series such as these, use the formula 

This works because you are taking the average of the largest and smallest terms, and then multiplying them by n, which is the same as calculating the sum total.

When k is not equal to 1, split the sums up into 2 piece and take the difference.

Find 

 

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