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 
.
http://matrix.skku.ac.kr/cal-lab/cal-10-2-16.html
2. 
11. 
12. 
.