Need help with this question:

True or false: if $(a_{n+1} -a_n)$ is a bounded sequence, then $\lim\limits_{n\mapsto \infty}(\frac{a_n}{n^2})=0$

I couldn't find counterexample and couldn't not prove it.

  • 1
    $\begingroup$ True by Cesàro-Stolz or by telescoping, of course. The exponent $2$ can be replaced by any exponent $>1$ and the claim still is true. $\endgroup$ – Jack D'Aurizio Nov 11 '17 at 14:04

Avail yourself of "telescoping."

$$a_n = a_0 + \sum_{k=1}^n (a_k - a_{k-1}).$$

  • $\begingroup$ $a_0$ is not defined as $n$ is a natural number $\endgroup$ – Avishay28 Nov 11 '17 at 13:18
  • $\begingroup$ Note that you can use $a_1$ and start the sum with $k=2$. $\endgroup$ – ncmathsadist Nov 11 '17 at 13:19

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