# GRE Subject Test: Math : Limits of Sequences

## Example Questions

### Example Question #2801 : Calculus Ii

There are 2 series,  and , and they are both convergent. Is  convergent, divergent, or inconclusive?

Convergent

Divergent

Inconclusive

Convergent

Explanation:

Infinite series can be added and subtracted with each other.

Since the 2 series are convergent, the sum of the convergent infinite series is also convergent.

Note: The starting value, in this case n=1, must be the same before adding infinite series together.

### Example Question #1 : Limits Of Sequences

You have a divergent series  , and you multiply it by a constant 10. Is the new series  convergent or divergent?

Inconclusive

Convergent

Divergent

Divergent

Explanation:

This is a fundamental property of series.

For any constant c, if  is convergent then  is convergent, and if  is divergent,  is divergent.

is divergent in the question, and the constant c is 10 in this case, so  is also divergent.

### Example Question #1 : Limits Of Sequences

There are 2 series and .

Is the sum of these 2 infinite series convergent, divergent, or inconclusive?

Convergent

Divergent

Inconclusive

Convergent

Explanation:

A way to find out if the sum of the 2 infinite series is convergent or not is to find out whether the individual infinite series are convergent or not.

Test the first series

.

This is a geometric series with .

By the geometric test, this series is convergent.

Test the second series

.

This is a geometric series with .

By the geometric test, this series is convergent.

Since both of the series are convergent,  is also convergent.

### Example Question #161 : Gre Subject Test: Math

Find the radius of convergence for the power series

Explanation:

We can use the limit

to find the radius of convergence. We have

This means the radius of convergence is .

### Example Question #2 : Ratio Test And Comparing Series

Determine if the following series is divergent, convergent or neither.

Divergent

Convergent

Inconclusive

Both

Neither

Divergent

Explanation:

In order to figure if

is convergent, divergent or neither, we need to use the ratio test.

Remember that the ratio test is as follows.

Suppose we have a series . We define,

Then if

, the series is absolutely convergent.

, the series is divergent.

, the series may be divergent, conditionally convergent, or absolutely convergent.

Now lets apply the ratio test to our problem.

Let

and

Now

.

Now lets simplify this expression to

.

Since ,

we have sufficient evidence to conclude that the series is divergent.

### Example Question #2 : Limits Of Sequences

Calculate the sum of the following infinite geometric series:

Explanation:

This is an infinite geometric series.

The sum of an infinite geometric series can be calculated with the following formula,

, where  is the first value of the summation, and r is the common ratio.

Solution:

Value of  can be found by setting

r is the value contained in the exponent

### Example Question #3 : Limits Of Sequences

Determine how many terms need to be added to approximate the following series within

Explanation:

This is an alternating series test.

In order to find the terms necessary to approximate the series within  first see if the series is convergent using the alternating series test. If the series converges, find n such that

Step 1:

An alternating series can be identified because terms in the series will “alternate” between + and –, because of

Note: Alternating Series Test can only show convergence. It cannot show divergence.

If the following 2 tests are true, the alternating series converges.

1.
2.        {} is a decreasing sequence, or in other words

Solution:

1.

2. {} is a decreasing functon, since a factorial never decreases.

Since the 2 tests pass, this series is convergent.

Step 2:

Plug in n values until

4 needs to be added to approximate the sum within .

### Example Question #1 : Limits Of Sequences

Evaluate: . (Round to 4 places)

Explanation:

Step 1: Plug in values into the function and add up the fraction:

Step 2: Find the sum of the fractions....

We can convert the fractions to decimals:

Step 3: Round to  places...