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Consider the series $\displaystyle{ \sum_{n=1}^{\infty} \left[ \frac{\sin \left( \frac{n^2+1}{n}x\right)}{\sqrt{n}}\left( 1+\frac{1}{n}\right)^n\right]}$ . Find all points at which the series is convergent. Find all intervals that the series is uniformly convergent.

I know that I need to use Dirichlet and/or Abel Criterion to show this.

My first attempt was to consider the argument of the sum as product of three sequences of functions, $f_n,g_n,h_n$ and perform Dirichlet/Abel twice. Towards that end I tried to break up $ \sin \left( \frac{n^2+1}{n}x\right)$ using $\sin(\alpha +\beta)=\sin \alpha \cos \beta \,+\, \sin \beta \cos \alpha $ to produce

$\sin \left(nx +\frac{x}{n} \right)=\sin(nx)\cos\left(\frac{x}{n}\right)+\sin\left(\frac{x}{n}\right)\cos (nx)$ and use the fact that $\sin(nx)\leq \frac{1}{\big|\sin\left(\frac{x}{2}\right)\big|}$, but then I don't know what to do with the $\sin\left(\frac{x}{n}\right)$, $\cos\left(\frac{x}{n}\right)$, and $\cos(nx)$. Previously, when using this method we showed uniform convergence on compact intervals like $[2k\pi-\varepsilon,2(k+1)\pi-\varepsilon]$.

Can I just say $ \frac{\sin \left( \frac{n^2+1}{n}x\right)}{\sqrt{n}}\leq \frac{1}{\sqrt{n}}\to 0$ as $n\to \infty$ ?

Also, I know that $\lim_{n\to \infty}\left( 1+\frac{1}{n}\right)^n=e$.

i.e. Help :) Thank you in advance

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2 Answers 2

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Note that

$$\sin \left( \frac{n^2+1}{n}x\right)=\sin \left( nx+\frac{1}{n}x\right)=\sin(nx)\cos\left(\frac{1}{n}x\right)+\cos(nx)\sin\left(\frac{1}{n}x\right)=$$

$$=\sin(nx)+\frac{x\cos(nx)}{n}+o\left(\frac1n\right)$$

$$\left( 1+\frac{1}{n}\right)^n=e^{n\log\left( 1+\frac{1}{n}\right)}=e^{1+\frac1n+o\left(\frac1n\right)}=e\left({1+\frac1n+o\left(\frac1n\right)}\right)$$

thus

$$\displaystyle{ \sum_{n=1}^{\infty} \left[ \frac{\sin \left( \frac{n^2+1}{n}x\right)}{\sqrt{n}}\left( 1+\frac{1}{n}\right)^n\right]}= e\sum_{n=1}^{\infty}\left[\frac{\sin(nx)}{\sqrt n}+\frac{x\cos(nx)}{n\sqrt n}+o\left(\frac1{n\sqrt n}\right)\right]$$

which converges since

$\sum_{n=1}^{\infty}\frac{\sin(nx)}{\sqrt n}$ converges by Dirichlet's test

$\sum_{n=1}^{\infty}\frac{x\cos(nx)}{n\sqrt n}$ converges absolutely by comparison with $\sum_{n=1}^{\infty}\frac{1}{n\sqrt n}$

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    $\begingroup$ Its a basic taylor's series application $$\cos\left(\frac{1}{n}x\right)=1+o(1)$$ $$\sin\left(\frac{1}{n}x\right)=\frac{1}{n}x+o\left(\frac{1}{n}x\right)$$ $\endgroup$
    – user
    Jan 7, 2018 at 9:06
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    $\begingroup$ @gimusi You're a little loose with the details here. And I think you've made a mistake: $(1+1/n)^n = e +o(1)$ is not enough to solve this. It leads to the series $$\sum_{n=1}^{\infty} \frac{\sin (nx)}{\sqrt n}o(1),$$ which can diverge. Fortunately $(1+1/n)^n = e +O(1/n),$ and this leads to a solution. $\endgroup$
    – zhw.
    Jan 7, 2018 at 19:59
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    $\begingroup$ @zhw. Thanks for the hint, indeed initially I expanded at $\left( 1+\frac{1}{n}\right)^n=e(1+1/n+o(1/n))$ and it seems not necessary then I went back to e+o(1). $\endgroup$
    – user
    Jan 7, 2018 at 20:08
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    $\begingroup$ @gimusi You're welcome. But no, it is not a minor issue. Your proof doesn't work using $(1+1/n)^n = e +o(1).$ Please check carefully. $\endgroup$
    – zhw.
    Jan 7, 2018 at 20:32
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    $\begingroup$ @zhw. I mean it is a minor issue in the sense that with the expansion at o(1/n) the susequent expression for the series doesn't change. Certainly its a gap in the proof and I had to better clarify thi point but it doesn't modifiy sustantially the solution as you noted. Thanks again for the clarification. $\endgroup$
    – user
    Jan 7, 2018 at 20:41
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Pointwise convergence: Fix $x.$ Our series equals

$$\tag 1 \sum_{n=1}^{\infty} \frac{\sin(nx+x/n)-\sin (nx)}{\sqrt n}(1+1/n)^n + \sum_{n=1}^{\infty} \frac{\sin (nx)}{\sqrt n}(1+1/n)^n.$$

Now by the MVT,

$$|\sin(nx+x/n)-\sin (nx)| = |(\cos c)(x/n)| \le |x/n|.$$

So in the first series in $(1),$ the sum of the absolute values is bounded above by

$$\sum_{n=1}^{\infty} \frac{|x/n|}{\sqrt n}(1+1/n)^n.$$

Because $(1+1/n)^n\le e,$ this series converges absolutely, hence converges.

In the second series in $(1),$ let's use $(1+1/n)^{1/n} = e+ O(1/n).$ (I'll leave this as an exercise.) Then this series equals

$$\sum_{n=1}^{\infty} \frac{\sin (nx)}{\sqrt n}e + \sum_{n=1}^{\infty}\sin (nx)\cdot O\left (\frac{1}{n^{3/2}}\right).$$

Here the the first series converges by Dirichlet's test, and the second series converges absolutely. Thus $(1)$ converges pointwise everywhere.

Uniform convergence: From the above, after dropping the constant $e,$ our series can be written

$$\tag 2 \sum_{n=1}^{\infty} \frac{\sin (nx)}{\sqrt n} + \sum f_n(x),$$

where $\sum f_n(x)$ converges uniformly on $[-r,r]$ for every $r>0.$ That's not true of the first series. This series is $2\pi$-periodic, so consider what happens on $[0,2\pi].$ Summing by parts (a la Dirichlet) shows that the series converges uniformly on $[a,2\pi-a]$ for all small $a>0.$

Suppose the convergence were uniform to some $f$ on $[0,2\pi].$ Then $f$ would be continuous and we would have

$$\int_0^{2\pi}|f|^2 = \lim_{N\to \infty} \int_0^{2\pi}\left |\sum_{n=1}^{N} \frac{\sin (nx)}{\sqrt n}\right |^2\, dx = \lim_{N\to \infty}\sum_{n=1}^{N} \frac{\pi}{(\sqrt n)^2} = \infty.$$

In the second $=$ we have unsed the orthogonality of the functions $\sin (nx).$ That's a contradiction, hence uniform convergence on $[0,2\pi]$ fails. Putting this all together shows the intervals of convergence for $(2)$ are $[2m\pi + a,2\pi(m+1)-a],m\in \mathbb Z,$ for any small $a>0.$

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  • $\begingroup$ @MarkViola Thank you and same back! $\endgroup$
    – zhw.
    Jan 7, 2018 at 19:52

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