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Mathematica tells the series below doesn't converge. I think it converges. What would the proper way to write things be as an input?

Sum[((-1)^(n) Log[1 + 2 n])/(1 + 2 n), {n, 0, Infinity}]
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By the alternating series test it's clear the series converges, but it seems the convergence is very slow. –  Chris's sis Apr 25 '13 at 18:54
    
What version of Mathematica are you using? It doesn't tell me that it does not converge. It simply can't calculate it. v9.0.1 here. –  Szabolcs Apr 25 '13 at 18:57
    
@Szabolcs I use v8.0 here. –  Chris's sis Apr 25 '13 at 18:58
    
Yes, I can reproduce that in v8. It seems it was a bug in v9, 'fixed' in a certain way in v9. No answer is better than a wrong answer. –  Szabolcs Apr 25 '13 at 18:59
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@belisarius We've been talking to ourself for days now... –  Daniel Lichtblau Apr 25 '13 at 21:38
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1 Answer

To get an actual answer in this case, you could simply retreat to the numerical evaluation of this sum. It works by simply changing Sum to NSum in your code:

NSum[((-1)^(n) Log[1 + 2 n])/(1 + 2 n), {n, 0, Infinity}]

(* ==> -0.192901 *)
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