The series can be 'splitted' as...
... and it becomes the sum of two series both converging for ...
Kind regards
Find the values for for which the following series converge:
1.
Attempted solution:
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.
2.
Attempted solution:
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.
You're correct, it converges for all values of .
But it doesn't mean that converges iff both and converge. For example for the first series does converge but the two others don't. It remains to be checked that diverges when or .
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 ?