# Calculus 2 : Types of Series

## Example Questions

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### Example Question #1 : Types Of Series

Evaluate:

The series diverges.

Explanation:

This can be rewritten as

, so , making this a convergent geometric series with initial term  and common ratio . The sum is therefore

.

### Example Question #2 : Types Of Series

Evaluate:

The series diverges.

Explanation:

This can be rewritten as

.

This is a geometric series with initial term  and common ratio . Since , and the series converges to:

### Example Question #3 : Types Of Series

Evaluate:

The series is not convergent.

Explanation:

is an infinite geometric series with initial term  and common ratio . The sum is therefore

### Example Question #4 : Types Of Series

Evaluate:

Explanation:

is a geometric series with initial term  and common ratio . The sum of this series is

.

### Example Question #5 : Types Of Series

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

Two

One

Three

Five

Four

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 #6 : Types Of Series

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

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 #7 : Types Of Series

What value does the series  approach?

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 #8 : Types Of Series

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

Series is not arithmetic

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 #9 : Types Of Series

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

Series does not converge.

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 #10 : Types Of Series

Calculate the sum of the following series:

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:

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