Interactive Real Analysis
- part of Examples 4.2.16(b):
The ratio test as well as the root test show that the series
converges.
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converges.
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We have to check the lim sup of the expression
Clearly, the lim sup of this expression is equal to 1/2. Hence, by the ratio test the series converges absolutely.
The root test for this series, of course, gives the same result. However, it is much easier to compute the limit of the above expression rather than computing it for the expression resulting in the root test.