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So I have this inequality and I just can't figure out how to prove it:

Prove that ($\forall n\in \mathbb N)$ $$\sum_{k=1}^n \frac{1}{(k+1)\sqrt k}<2.$$ I've figured that for $n=1$ the inequality holds, since: $\frac{1}{2}<2$; so the statement is true for some $n \in \mathbb N$. Although I can't figure out how to prove the implication, that if the statement holds for some $n \in \mathbb N$, then it also holds for the number $n+1$.

I can't seem to make use of the induction hypothesis since the right-hand side is just a constant.

I'd be grateful for any suggestions.

edit: Solved, thanks for the answers.

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  • $\begingroup$ Compare it with a suitable integral? It's the first thing I would look to, at least. $\endgroup$ – Harald Hanche-Olsen Jan 8 '14 at 21:05
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    $\begingroup$ Find a sequence $c_n > 0$ such that $$\sum_{n=1}^N \frac{1}{(n+1)\sqrt{n}} < 2 - c_N.$$ $\endgroup$ – Daniel Fischer Jan 8 '14 at 21:05
  • $\begingroup$ Looks like $n$ in the series is a dummy index.. But you don't refer to it as such.. Can you clarify? $\endgroup$ – user76568 Jan 8 '14 at 21:07
  • $\begingroup$ I'm sorry, seems that I made a mistake, what I meant is that it holds for any $n \in \mathbb N$ from $k=1$ to $n$, thanks for noticing. I've changed it in the post. $\endgroup$ – David Jan 8 '14 at 21:16
  • $\begingroup$ @David : this problem is equivalent to showing $\sum_{k=1}^\infty \frac{1}{(k+1)\sqrt{k}} \leq 2$. $\endgroup$ – Stefan Smith Jan 9 '14 at 2:38
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Note that

$$\frac1{\sqrt{k}}-\frac1{\sqrt{k+1}}=\frac1{\sqrt{k}\sqrt{k+1}(\sqrt{k}+\sqrt{k+1})}\ge\frac1{2(k+1)\sqrt{k}}.$$

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  • $\begingroup$ So the LHS is basically a telescoping series, which converges to 1 and after multiplying the inequality by 2 I get the desired result, right? Actually the "$\ge$" sign can be changed to a "$>$" sign since it holds for every k, to fit the original problem. I just wanted to ask about the equality on the left side, could you elaborate on that a bit? Thanks a lot, by the way. $\endgroup$ – David Jan 9 '14 at 21:36
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    $\begingroup$ Nveremind, I figured out: $\frac1{\sqrt{k}}-\frac1{\sqrt{k+1}}=\frac{\sqrt{k+1}-\sqrt k}{\sqrt k \sqrt{k+1}}=\frac{(\sqrt{k+1}-\sqrt k)(\sqrt{k+1}+\sqrt k)}{(\sqrt k \sqrt{k+1})(\sqrt{k+1}+\sqrt k)}=\frac1{\sqrt{k}\sqrt{k+1}(\sqrt{k}+\sqrt{k+1})}$. $\endgroup$ – David Jan 9 '14 at 21:49
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Estimate the $\sum_{n=N}^\infty$ by the integral of the function (from $N$ to $\infty$). Notice that your function is decreasing and convex, so you have two-sided bounds.

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