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Question Number 64671 by aliesam last updated on 20/Jul/19

Commented by mathmax by abdo last updated on 20/Jul/19

$$wehavex-\frac{x^3}{6}⩽sinx⩽xforallxreal\Rightarrow$$$$n^{2}x-\frac{n^6{x}^3}{6}⩽sin\left(n^{2}x\right)⩽n^{2}x\Rightarrow nx-\frac{n^5}{6}{x}^3⩽\frac{sin\left(n^{2}x\right)}{n}⩽nx\Rightarrow$$$$\sum _{n=1}^N(nx-\frac{n^5}{6}{x}^3)⩽\sum _{n=1}^N\frac{sin\left(n^{2}x\right))}{n}⩽\sum _{n=1}^{N}nx\Rightarrow$$$$\frac{N(N+1)}{2}x-x^3\sum _{n=1}^{\mathrm{\infty }}\frac{n^5}{6}⩽\sum _{n=1}^N\frac{sin\left(n^{2}x\right)}{n}⩽\frac{N(N+1)}{2}x\mathrm{\forall }N\in N\Rightarrow$$$$(x\to 0)\Rightarrow {lim}_{x\to 0}\sum _{n=1}^N\frac{sin\left(n^{2}x\right)}{n}=0forallN\Rightarrow$$$${lim}_{x\to 0}\sum _{n=1}^{\mathrm{\infty }}\frac{sin\left(n^{2}x\right)}{n}=0.$$

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