9.3 Alternating Series and Absolute Convergence
by SGLee - HSKim - SWSun
1-7. Test for convergence of the following alternating series:
1. 
Since 
is divergent, the series is divergent by the Integral Test.
Calculus-Sec-9-3-Solutio
9.3 Alternating Series and Absolute Convergence
by SGLee - HSKim - SWSun
1-7. Test for convergence of the following alternating series:
1.
Since is divergent, the series is divergent by the Integral Test.
2. 
Note that 
iff 
, 
By Alternating Series Test, the series is divergent.
3. 
It satisfies 
because 
and 
.
By the Alternating Series Test, the series is convergent.
5. 
By the Integral Test,
is divergent.
By Alternating Series Test, the series is divergent.
6. 
By the Ratio Test, this series is absolutely convergent.
Hence, 
is convergent.
7. 
By Alternating Series Test, the series is divergent.
8. For what values of 
, the series 
is convergent.
By Exercise 5, when 
, this series diverges.
Let 
and 
.
For 
, 
for all x.
And for 
, 
Thus, by the Alternating Series Test, for 
, this series is convergent. If 
, this series is divergent.
9-14. Test whether the series is absolutely convergent, conditionally convergent, or divergent.
9. 
Hence, the series is divergent.
11. 
.
Since 
for all 
, we have 
.
Thus 
Let 
is absolutely convergent.
By the Comparison Test,
is absolutely convergent, and hence convergent.
12. 
Using the root test,
is absolutely convergent, and hence convergent.
13. 
.
We do not know if this series is absolutely convergent, when using the ratio test. Let us try another test.
.
Here, 
for all 
and 
By the Alternating Series Test,
is conditionally convergent.
14. 
Then
This means we cannot conclude convergence of this series using the Ratio Test.
Consider 
Here, 
for all 
and 
By the Alternating Series Test,
is conditionally convergent.
4. 
10. 
