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)

  • $\begingroup$ 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 $0$. $\endgroup$ Mar 27, 2012 at 5:34
  • $\begingroup$ Oh god dammit, I was thinking of a limit. Nevermind. $\endgroup$ Mar 27, 2012 at 5:36
  • $\begingroup$ $\log_e(4)-1$ if you really want to know $\endgroup$
    – Henry
    Mar 27, 2012 at 6:40

1 Answer 1


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 $$\dfrac{1}{2^n(n + 1)} < \dfrac{1}{2^n}$$

  • 2
    $\begingroup$ Thank you... :-) (Now wait for someone to comment that this is a strange comment :-)). $\endgroup$
    – Aryabhata
    Mar 27, 2012 at 6:14
  • $\begingroup$ (My comments condensed into an answer) $\endgroup$ Mar 27, 2012 at 6:15
  • 3
    $\begingroup$ @Aryabhata: This is a strange comment :) $\endgroup$
    – t.b.
    Mar 27, 2012 at 6:34
  • $\begingroup$ @t.b.: ....... :-) $\endgroup$
    – Aryabhata
    Mar 27, 2012 at 11:59

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