The series can be 'splitted' as...
... and it becomes the sum of two series both converging for ...
Find the values for for which the following series converge:
I tried to use the root test to evaluate and found that it is equal to . I got stuck here so I tried to use the ratio test and got to . And I'm stuck here for this method as well.
I thought the only useful test would be the ratio test. So I tried it and got
however then I found which implies all values make this series converge, but that doesn't seem right.
Any suggestions would be appreciated. Thanks in advance.
I think that using the ratio test is simpler.
If , hence
and as , the series converges. Similarly one can show that if the series diverges. To check if the series converges when simply substitute or :
- If ,Is that a convergent series ?
- If ,because when is even. Does this series converge ?