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.