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| PAGE: 770 | SET: Select | PROBLEM: 21 |
The series is 
.
Ratio test :
Let 
be a series with non zero terms.
1. converges absolutely if 
.
2. diverges if 
or 
.
3. The ratio test is inconclusive if 
.
Here 
and 
.
Find 
.
By the ratio test, the series 
is convergent when 
.
Radius of the convergence is half the width of the interval.
Radius of convergence is 
.
Check the interval of convergence at the end points.
For 
, 
.
Divergence test:
If 
does not exist or if 
, then the series 
is divergent.
.
The series is divergent by divergence test.
The series 
is convergent by alternating series test.
For 
, 
is also divergent.
Therefore, interval of convergence is 
.
Radius of convergence is 
.
Interval of convergence is 
.
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