<s>[INST] the series $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\sin(1+\frac{x}{n})}$ converges uniformly in $[-a,a]$ I have to show that the series $\displaystyle{\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}\sin(1+\frac{x}{n})}$ converges uniformly in $[-a,a], a>0$.
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That's what I have tried:
$w_n=\frac{(-1)^n}{\sqrt{n}} \sin1 \cos(\frac{x}{n})+\frac{(-1)^n}{\sqrt{n}} \cos1 \sin(\frac{x}{n})=\frac{(-1)^n}{\sqrt{n}} \sin1 (\cos(\frac{x}{n})+1-1)+\frac{(-1)^n}{\sqrt{n}} \cos1 \sin(\frac{x}{n})= \frac{(-1)^n}{\sqrt{n}} \sin1 (\cos(\frac{x}{n})-1)+\frac{(-1)^n}{\sqrt{n}} \sin1 +\frac{(-1)^n}{\sqrt{n}}  \cos1 \sin(\frac{x}{n})=\sin1 \cdot g_n(x)+ \cos1 \cdot  f_n(x)+\frac{(-1)^n}{\sqrt{n}} \sin1 $ .
We set $d_n(x)=w_n-\frac{(-1)^n}{\sqrt{n}} \sin1=\sin1 \cdot g_n(x)+ \cos1 \cdot  f_n(x)$
$|g_n(x)|=|\frac{(-1)^n}{\sqrt{n}}(\cos(\frac{x}{n})-1)|=\frac{1}{\sqrt{n}} (\cos(\frac{x}{n})-1) (\cos(\frac{x}{n})+1)=\frac{1}{\sqrt{n}}(\sin^2(\frac{x}{n})) \leq \frac{1}{\sqrt{n}} \cdot \frac{x^2}{n^2} \leq \frac{1}{\sqrt{n}} \cdot \frac{a^2}{n^2}=\frac{a^2}{n^{\frac{5}{2}}}$    
$\sum_{n=1}^{\infty} \frac{a^2}{n^{\frac{5}{2}}}< +\infty$.
$|f_n(x)|=|\frac{(-1)^n}{\sqrt{n}} \sin(\frac{x}{n})| \leq \frac{1}{\sqrt{n}} \frac{x}{n} \leq \frac{a}{n^{\frac{3}{2}}}$
$\sum_{n=1}^{\infty} \frac{a}{n^{\frac{3}{2}}}< +\infty$.
Also,from Dirichlet's test,we know that $\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$ converges,let $b:=\sum_{n=1}^{\infty} \frac{(-1)^n}{\sqrt{n}}$
Therefore,we have $d_n(x) \leq \sin1 \frac{a^2}{n^{\frac{5}{2}}}+ \cos1  \frac{a}{n^{\frac{3}{2}}}$.
$\sum_{n=1}^{\infty} \sin1 \frac{a^2}{n^{\frac{5}{2}}}+ \cos1  \frac{a}{n^{\frac{3}{2}}}$ converges,as $\sum_{n=1}^{\infty} \frac{a^2}{n^{\frac{5}{2}}}$ converges  and $\sum_{n=1}^{\infty} \frac{a}{n^{\frac{3}{2}}}$ converges too.
So, $\sum_{n=1}^{\infty} d_n(x)$ converges uniformly to a $s(x)$.
$\sum_{n=1}^{\infty} d_n(x)=\sum_{n=1}^{\infty}  (w_n-\frac{(-1)^n}{\sqrt{n}} \sin1)= \sum_{n=1}^{\infty} w_n-b \overset{uniformly}{=}s(x) \Rightarrow \sum_{n=1}^{\infty} w_n=b+s(x):=q(x):[-a,a] $
So,the series converges uniformly to $q(x)$.
Could you tell me if it is right?
 [/INST] Hint You may use Dirichlet's criterion, since $\sin (xn^{-1})$ is eventually decreasing in $[0,M]$ and $1-\cos (xn^{-1})$ is eventually decreasing in $[0,M]$ too, and both go uniformly to zero, and $$\sum_{n\geqslant 1} (-1)^{n}n^{-1/2}$$ converges. 
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