The Fourier series of $f(x)=|x|$ on $[-\pi,\pi]$ can be written as $$f(x)\sim \dfrac{\pi}{2}+2\sum_{n=1}^{\infty}\dfrac{(-1)^{n}-1}{\pi n^{2}}e^{-inx}.$$
This post Show absolute and uniform convergence of a Fourier series has shown that this Fourier series converges uniformly on $[-\pi,\pi]$. However, I want to show that this Fourier series converges uniformly to $f$ on the whole $[-\pi,\pi]$.
To this end, I define the partial sum $$S_{N}:=\dfrac{\pi}{2}+2\sum_{n=1}^{N}\dfrac{(-1)^{n}-1}{\pi n^{2}}e^{-inx},$$ and then try to estimate $|f(x)-S_{N}(x)|$. I have some preliminary result, but it does not give me $$\sup_{x\in[-\pi,\pi]}|f(x)-S_{N}(x)|\longrightarrow 0,$$ when $N\rightarrow\infty$. Certainly, I can use the $\epsilon-N$ definition, but my estimate outcome is not that computation friendly either for me to get $N$ for each $\epsilon$.
The following is my estimate:
As $|e^{-inx}|=1$, we can have the following estimate
\begin{align*}
|f(x)-S_{N}(f)(x)|=\Bigg|f(x)-\dfrac{\pi}{2}-2\sum_{n=1}^{N}\dfrac{(-1)^{n}-1}{\pi n^{2}}e^{-inx}\Bigg|&\leq |f(x)|+\dfrac{\pi}{2}+2\sum_{n=1}^{N}\Bigg|\dfrac{(-1)^{n}-1}{\pi n^{2}}\Bigg|\\
&=|x|+\dfrac{\pi}{2}+2\sum_{n=1}^{N}\Bigg|\dfrac{(-1)^{n}-1}{\pi n^{2}}\Bigg|\\
&\leq \dfrac{3\pi}{2}+\dfrac{4}{\pi}\sum_{n=1}^{N}\dfrac{1}{n^{2}}\\
&\leq \dfrac{3\pi}{2}+\dfrac{4}{\pi}\Bigg(\dfrac{\pi^{2}}{6}-\dfrac{1}{N+1}\Bigg)\\
&=\dfrac{3\pi}{2}+\dfrac{2\pi}{3}-\dfrac{4}{\pi(N+1)}.
\end{align*}
Then I got stuck. We have a good outcome that the bound does not depend on $x$, but solve the bound $<\epsilon$ seems really complicated. Is there any way for me to make this nicer? The best case would be that I can directly conclude the bound goes to $0$ when $N\rightarrow\infty$.
Thank you!
Edit: Pointwise Convergence
As the answer of "Mostafa Ayaz" suggested, we need to firstly prove that the Fourier series converges to $f(x)$ pointwise on the interval $[-\pi,\pi]$.
In fact, the reason why I directly proved the uniform convergence was that I did not know how to prove the pointwise convergence.
I mean, it is straightforward to prove that the series convergences, but how to prove it convergent pointwise to $f(x)$ on the whole $[-\pi,\pi]$?
Edit 2:
Okay. I just recalled that $f(x)=|x|$ is Holder continuous, so the partial sum must converge pointwise.
Best Answer
Your approach is correct, but using the triangle inequality at the first step is a bit overkill and leads to irrelevant bounds. Just do the following $$ |f(x)-S_{N}(f)(x)|{=\Bigg|f(x)-\dfrac{\pi}{2}-2\sum_{n=1}^{N}\dfrac{(-1)^{n}-1}{\pi n^{2}}e^{-inx}\Bigg| \\= \Bigg|\dfrac{\pi}{2}+2\sum_{n=1}^{\infty}\dfrac{(-1)^{n}-1}{\pi n^{2}}e^{-inx}-\dfrac{\pi}{2}-2\sum_{n=1}^{N}\dfrac{(-1)^{n}-1}{\pi n^{2}}e^{-inx}\Bigg| \\= \Bigg|2\sum_{n=N+1}^{\infty}\dfrac{(-1)^{n}-1}{\pi n^{2}}e^{-inx}\Bigg| \\\le \Bigg|2\sum_{n=N+1}^{\infty}\dfrac{(-1)^{n}-1}{\pi n^{2}}\Bigg| \\\le \sum_{n=N+1}^{\infty}\dfrac{4}{\pi n^{2}} } $$ From now, it is very straightforward.