2
$\begingroup$

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)

$\endgroup$
3
  • $\begingroup$ 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 $0$. $\endgroup$
    – Samuel Reid
    Mar 27, 2012 at 5:34
  • $\begingroup$ Oh god dammit, I was thinking of a limit. Nevermind. $\endgroup$
    – Samuel Reid
    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

Reset to default
9
$\begingroup$

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

$\endgroup$
4
  • 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) $\endgroup$
    – The Chaz 2.0
    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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.