# Convergence and Divergence

## Example Questions

### Example Question #2951 : Calculus Ii

Determine the convergence of the series using the Comparison Test.

Series converges

Series diverges

Cannot be determined

Series diverges

We compare this series to the series

Because

for

it follows that

for

This implies

Because the series on the right has a degree of equal to in the denominator,

the series on the right diverges

making

diverge as well.

### Example Question #2952 : Calculus Ii

Does the series converge or diverge? If it does converge, then what value does it converge to?

Diverges

Converges to 1

Converges to

Converges to

Converges to

Converges to 1

To show this series converges, we use direct comparison with

,

which converges by the p-series test with .

Thus we must show that

.

Cross multiplying the previous section and multiplying by , we obtain .

Since this holds for all we can conclude that

.

Summing from to , and noting that

for all
, we obtain the following inequality:

.

Therefore the series

converges by direct comparison.

Now to find the value, we note that

,

so that

.

Now let

be a sequence of partial sums.

Then we have

Therefore

.

Taking the limit as , we obtain the following:

Therefore we have

.

### Example Question #2953 : Calculus Ii

Does the series converge?

Cannot be determined

Yes

No

No

Notice that for

This implies that

for

Which then implies

Since the right-hand side is the harmonic series, we have

and thus the series does NOT converge.

### Example Question #2954 : Calculus Ii

Determine whether the series converges, absolutely, conditionally or in an interval.

Converges absolutely

Does not converge at all

Converges in an interval

Converges conditionally

Converges absolutely

### Example Question #2955 : Calculus Ii

Determine whether the series converges

Converges in an interval

Does not converge at all

Converges absolutely

Converges conditionally

Converges absolutely

### Example Question #2956 : Calculus Ii

Test for convergence

Converges conditionally

Cannot be determined

Converges in an interval

Converges absolutely

Diverges

Converges absolutely

Step 1: Recall the convergence rule of the power series:

According to the convergence rule of the power series….

converges as long as

Step 2: Compare the exponent:

Since , it is greater than . Hence the series converges.
Step 3: Conclusion of the convergence rule

Now, notice that the series isn’t an alternating series, so it doesn’t matter whether we check for absolute or conditional convergence.

### Example Question #2957 : Calculus Ii

Test for convergence

Can’t be determined

Diverges

Converges absolutely

Converges Conditionally

Converges in an interval

Converges absolutely

Step 1: Try and look for another function that is similar to the original function:

looks like

Step 2: We will now Use the Limit Comparison test

Since the limit calculated, is not equal to 0, the given series converges by limit comparison test

### Example Question #2958 : Calculus Ii

Does the following series converge or diverge?

Absolutely converge

Conditionally converge

Diverge

The series either absolutely converges, conditionally converges, or diverges.

Absolutely converge

The best way to answer this question would be by comparing the series to another series,, that greatly resembles the behavior of the original series, . The behavior is determined by the terms of the numerator and the denominator that approach infinity at the quickest rate. In this case:

When this series is simplifies, it simplifies to a series that converges because of the p-test where .

With two series and the confirmed convergence of one of those series, the limit comparison test can be applied to test for the convergence or divergence of the original series. The limit comparison test states that two series will converge or diverge together if:

Specifically:

This limit equals one because of the fact that:

if the coefficients come from the same power.

Because the limit is larger than zero, and will converge or diverge together. Since it was already established that converges, the original seies, , converges by the limit comparison test.

### Example Question #2959 : Calculus Ii

Determine if the following series converges or diverges:

Series diverges

Series converges

Series converges

for all
;
is a sum of geometric sequence with base 1/3.

Therefore, said sum converges.

Then, by comparison test, also converges.

### Example Question #2960 : Calculus Ii

What can be said about the convergence of the series ?

Diverges

Inconclusive

Converges

Converges

Since for all n>0, and converges as a P-Series, we may conclude that must also converge by the comparison test.

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