I would like to find out the sum of the following series $$ \lim_{n\to\infty} \left[\frac{n^2}{{(n^2+1)}^{3/2}} + \frac{n^2}{{(n^2+2)}^{3/2}} + \dots + \frac{n^2}{{(n^2+(n+1)^2)}^{3/2}}\right] $$ now it would be just upto $n$ then I could have calculate with Riemann's integral by narrowing it and simplify then just integrate the definite integration. As it's upto $n+1$, I am having the trouble simplifying it. Any help?

  • $\begingroup$ I think there is a mistake in your calculation. How did you reach to this sum? $\endgroup$ – Mhenni Benghorbal Jun 21 '13 at 11:14

There are $n+1$ terms, and all but the last are at least $1/2$. So...

  • $\begingroup$ What is the limit of $\dfrac{n+1}{2}$ as $n \to \infty$? $\endgroup$ – Robert Israel Jun 21 '13 at 11:02
  • $\begingroup$ But maybe the real question was supposed to have $n$ instead of $n^2$ in the numerator. $\endgroup$ – Robert Israel Jun 21 '13 at 11:02
  • 2
    $\begingroup$ @RobertIsrael, there are $(n+1)^2$ terms, as I see. $\endgroup$ – Oleg567 Jun 21 '13 at 11:06
  • $\begingroup$ Good point. So to get a nontrivial limit, you want $1$ in the numerator. $\endgroup$ – Robert Israel Jun 21 '13 at 12:23
  • 1
    $\begingroup$ @TamimAdDari,but 2nd one is 2 or 4? $\endgroup$ – chenbai Jun 23 '13 at 9:21

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