10.2 Tests for convergence of series with positive terms

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

1. .

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

.

Hence by the Integral Test, is diverges.

2. .

http://matrix.skku.ac.kr/cal-lab/cal-10-2-2.html

Sol)

var('x') f=x*exp(-x) integral(f, x, 1, oo) |

2*e^(-1) (converge to )

Thus, the series converges.

3. .

Sol) The function is continuous, positive on , and also decreasing. hence by Integral test, since for , so we can use the Integral Test.

.

Hence by Integral Test diverges.

4. .

Sol) The function is continuous, positive on , and also decreasing, since if , so we can use the Integral Test.

.

Hence by the Integral Test, converges.

5.

Sol)

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.

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

6. .

Sol) 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. .

Sol) We have already shown (in Exercises 3) 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.

Thus the series converges for .

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

Sol) 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. .

Sol) , so the series converges by comparison with the -series

10. .

Sol) Use the Limit Comparison Test with and .

Since converges, converges.

11. .

http://matrix.skku.ac.kr/cal-lab/cal-10-2-11.html

Sol)

var('n') u(n)=2^n*factorial(n)/n^n limit(u(n+1)/u(n), n=+oo) |

2*e^(-1)

bool(2*e^(-1)<1) |

True

12. .

http://matrix.skku.ac.kr/cal-lab/cal-10-2-12.html

Sol)

var('n') u(n)=n^2/3^n limit(u(n+1)/u(n), n=+oo) |

1/3

13. .

Sol) 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?

Sol) Use the Comparison Test.

If for all , then is convergent.

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

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

Sol) 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.