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)

  • 0
    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 $0$.2012-03-27
  • 0
    Oh god dammit, I was thinking of a limit. Nevermind.2012-03-27
  • 0
    $\log_e(4)-1$ if you really want to know2012-03-27

1 Answers 1

9

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
    Thank you... :-) (Now wait for someone to comment that this is a strange comment :-)).2012-03-27
  • 0
    (My comments condensed into an answer)2012-03-27
  • 3
    @Aryabhata: This is a strange comment :)2012-03-27
  • 0
    @t.b.: ....... :-)2012-03-27