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Hi I really don’t know how to solve this question:

For which a do these series converge ?
$(\sum_{i=0}^n \frac{1}{i!})^a $ only if i is odd
$\sum_{i=0}^n \frac{(2^i)}{i!}$ only if i is even

I know the even serie converge to something smaller than e^2 But for the odd one I don’t know how to handle it.

Any idea ? Thanks

Jouja
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  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Jan 03 '22 at 09:35
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    I’m a little confused by the question. The first expression is not so much a series as it is a power of a series. Is it correct? Or should it perhaps be $\sum_{i=0}^n\frac{1}{(i!)^a}$? The second expression is a series, but there is a suspicious lack of an $a$. Was there supposed to be an $a$ somewhere? – Theo Bendit Jan 03 '22 at 09:36
  • The first series is bounded by $e^a$ , hence converges for every real $a$ – Peter Jan 03 '22 at 09:37

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