Calculus-Sec-9-2-Solution

9.2  Tests for Convergence of Series         by SGLee - HSKim-SWSun

1-5. Determine whether the series is convergent or divergent using the Integral Test.

1.

The function  is continuous, positive, and decreasing on . The Integral Test applies.

.

Hence by the Integral Test,   is diverges.

Thus, the series converges.

3.

is continuous and positive on ,

and also decreasing since  for .

Hence we can use the Integral Test.

.

Thus,  converges. And by the Comparison Test,  is also convergent.

4.

The function  is continuous, positive on , and also decreasing.

Then, by Integral test, since  for , so we can use the Integral Test.

.

Hence by Integral Test  diverges.

5.

The function  is continuous, positive on ,

and also decreasing, since   if  ,

so we can use the Integral Test and .

.

Hence by the Integral Test,  converges.

6-7. Find the values of  for which the series is convergent.

6.

When  is continuous and positive on ,

and also decreasing since  for  (  for ).

Hence we can use the Integral Test.

Thus, when  diverges.

When  is continuous and positive on ,

and also decreasing since  for ,

so we can use the Integral Test,

This limit exists whenever , so the series converges for .

7.

We have already shown (in Exercise 4) that when , the given series is divergent.

Let us assume .

The function  is continuous, positive on ,

and also decreasing since  if ,

so we can use the Integral Test and .

Thus the series converges for .

8. Find all positive values of  for which the series  converges.

The function  is continuous, positive on ,

and also decreasing since  if  , so we can use the Integral Test:

Thus   for .  The series converges for

9-13. Test for convergence or divergence of the series.

9.

, so the series  converges by comparison with the -series .

10.

Use the Limit Comparison Test with  and .

Since  converges,   converges.

13.

Use the Limit Comparison Test with  and .

Then  and  are series with positive terms and

.

Since  converges, converges.

14. If  is a convergent series with non-negative terms, is it true that  is also convergent?

Use the Comparison Test.

If  for all , then  is convergent.

But, if  for all  then we cannot decide the convergence of .

15. If  is a convergent series with positive terms, is it true that  is also convergent?

Yes. Since  is a convergent series with positive terms, ,

and  is a series with positive terms (for large enough ).

We have .

Thus,  is also convergent by the Limit Comparison Test.

16. Find the radius of convergence of .

17. Find the radius of convergence of .