# Tricky Complex Series

by matt.qmar
Tags: complex, series, tricky
 P: 8 Hey, I am trying to determine the convergence/divergence of Ʃn=1∞ in/n. I have tried all the tests I could think of (Comparison, Ratio, Root, nth term) and cannot determine it's convergence. If there was a formula, for say, the Mth partial sum SM then, if the limit as M → ∞ of SM is L, we have convergence to L but I can't seem to arrange for the thing to add up the first M terms. Clearly, I think something can be done with the fact that in = {i, -1, -i, 1} repeatably with periodicity 4. I'm not sure how this can exactly be of help though! Any help appreciated, thanks.
 P: 1,649 Do you know the power series for $\log(1-z)$? It has a raidus of convergence equal to one. However convergence on the boundary $|z|=1$ is well, what? Take a look at Wikipedia for "Radius of Convergence".
 P: 8 Ah! I see, and we have |i| = 1 but i ≠ 1 so there we go! Thanks a bunch. Factoring out the -1 seems to make it all go quite nicely.
P: 8

## Tricky Complex Series

Ʃn=1 in/n = -log(1-i), woo hoo!
 P: 1,649 You know, that's really not good enough. Why does it converge on the boundary except for z=1? Or rather prove that it does. If you wish anyway.
 P: 8 Ah, Thanks for the heads up. I suppose I sort of took that for granted. Would http://en.wikipedia.org/wiki/Abel%27...mplex_analysis works to show convergence for all z ≠ 1 on the boundary (in particular, at our point i ≠ 1), as the an's are monotonically decreasing? Some the an, however, are negative... Trying to take some zo ≠ 1 with |zo| = 1 and using some other test (root, comparison, etc.) doesn't seem to work either, though...
 P: 1,649 Yeah, I don't know off-hand how to show that. There are tests to check for convergence on the boundary. Would need to look into thoses. Don't thinik Abel test would apply though.

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