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Quite an easy question, but I can't do it. I've even tried the tests I knew wouldn't work, (integral test, etc.) and I don't know what to do.

Determine if $\sum\limits_{n=1}^{\infty} \frac{1}{2^n (n+1)}$ converges or diverges.

I suspect it converges but I am not sure. (Not homework, just doing practice questions)

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Care to give me a hint if you see it? I know it is really easy... anything I tried to compare it with I would just get the limit equals $0$. –  Samuel Reid Mar 27 '12 at 5:34
    
Oh god dammit, I was thinking of a limit. Nevermind. –  Samuel Reid Mar 27 '12 at 5:36
    
$\log_e(4)-1$ if you really want to know –  Henry Mar 27 '12 at 6:40
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1 Answer

up vote 9 down vote accepted

We compare with $\dfrac{1}{2^n}$, the sum of which converges. Since $2^n(n + 1) > 2^n$ for $n \geq 1$, we have $$\dfrac{1}{2^n(n + 1)} < \dfrac{1}{2^n}$$

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Thank you... :-) (Now wait for someone to comment that this is a strange comment :-)). –  Aryabhata Mar 27 '12 at 6:14
    
(My comments condensed into an answer) –  The Chaz 2.0 Mar 27 '12 at 6:15
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@Aryabhata: This is a strange comment :) –  t.b. Mar 27 '12 at 6:34
    
@t.b.: ....... :-) –  Aryabhata Mar 27 '12 at 11:59
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