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Does the series $\sum_{n=1}^{\infty}((1/n)-\sin(1/n))$ converge..? Can anyone please give me a simple proof..

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Just use Taylor expansion for $\sin (1/n)$ as $n \to \infty$ – Cortizol Feb 26 '13 at 20:07
up vote 8 down vote accepted

Consider that

$$\frac{1}{n} - \sin{\left(\frac{1}{n}\right)} \sim \frac{1}{6 n^3}$$

as $n \rightarrow \infty$. Then use the comparison test.

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In case you don't know the power series but do know that $\sin'(t) = \cos(t)$ and $\sin(t) \le t$ for $t \ge 0$:

$x-\sin(x) = \int_0^x (1-\cos(t))dt$.

$\cos(2t) = \cos^2(t)-\sin^2(t) = 1-2\sin^2(t)$ so $1-\cos(t) = 2\sin^2(t/2) \le t^2/2$ since $|\sin(t)| \le |t|$.

Therefore $x-\sin(x) \le \int_0^x (t^2/2)dt = x^3/6$ and the remainder of the proof goes through as before.

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Yet you can show the convergence of $\sum_{n=1}^{\infty} \left(\frac{1}{n}-\sin\left(\frac{1}{n}\right)\right)$ as follows, which is based just on the inequality \begin{gather*}\sin(x)<x,\qquad \forall x>0.\end{gather*}

Indeed, since for all $n\in\mathbb{N},$ $1/n>0,$ we arrive at $0<1/n-\sin(1/n),$ and so the series under consideration is of strictly positive terms. Now we show that \begin{gather*}\tag{$\star$}x-\sin(x)<x^2, \qquad \forall x\in (0,1].\end{gather*} To this end, introduce the auxiliary function $f: x\mapsto x^2-x+\sin(x).$ Then $f$ is certainly continuous on the interval $[0,1].$ Note that \begin{align*} f'(x)&=2x-1+\cos(x)=2x-\big(1-\cos(x)\big)\\ &=2x-2\sin^2\left(\frac{x}{2}\right)\\ &>2x-2\cdot\left(\frac{x}{2}\right)^2\\ &=\frac{x}{2}(4-x)>0,\qquad \forall x\in(0,1), \end{align*} and thus we conclude that $f$ is increasing strictly on $[0,1].$ As a result, $$x^2-x+\sin(x)=f(x)>f(0)=0,\qquad \forall x\in (0,1].$$ Hence $(\star)$ is valid.

Finally, by $(\star),$ we get \begin{gather*} 0<\frac{1}{n}-\sin\left(\frac{1}{n}\right)<\frac{1}{n^2}, \qquad \forall n\in\mathbb{N}. \end{gather*} Since $\sum_{n=1}^{\infty} 1/n^2$ converges, we deduce that $\sum_{n=1}^{\infty} (1/n-\sin(1/n))$ is convergent, by comparison test.

Furthermore, if you feel the above argument is too clumsy, try as follows. It is easy to show that (for instance, by L'Hospital's Rule) \begin{gather*}\lim_{x\to 0} \frac{x-\sin(x)}{x^2}=0<1,\end{gather*} and then, by the locally order-preserving property of limit, there exists some $\delta>0$ such that \begin{gather*} x-\sin(x)<x^2,\qquad \forall x\in (0, \delta).\end{gather*} Therefore, there exists some natural number $N$ such that $1/N<\delta.$ Hence \begin{gather*} \frac{1}{n}-\sin\frac{1}{n}<\frac{1}{n^2}, \qquad \forall n>N.\end{gather*} Then the desired result follows.

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