<s>[INST] Fourier serie for specific value (follow up question) This is a follow-up question to this.
I now understand that when calculating a specific value in a Fourier series, the goal is to identify a pattern and recognize a known infinite series or Taylor expansion.
I have the function $g(x) = 5x+7$, and its Fourier series: $\displaystyle7 + \sum_{n=1}^{\infty} \frac{-1^{n+1} \cdot 80}{n \pi} \sin \left(\frac{n \pi x}{8}\right)$. I verified it was correct. I now need to find f(5).
As the teacher showed me, I took several values of $n$ for $\sin(\frac{5n\pi}{8})$ to try to identify my infinite sum.
$$
\begin{aligned}
\displaystyle
\begin{array}{lll}
\displaystyle
n=1 & \sin \left(\frac{5 \pi}{8}\right) & \frac{\sqrt{2+\sqrt{2}}}{2} \\
n=2 & \sin \left(\frac{5 \pi}{4} \right) & \frac{-1}{\sqrt{2}} \\
n=3 & \sin \left(\frac{15 \pi}{8} \pi\right)=\sin \left(\frac{- \pi}{8}\right) & -\frac{\sqrt{2-\sqrt{2}}}{2}\\
n=4 & \sin \left(\frac{5 \pi}{2} \pi\right)=\sin \left(\frac{\pi}{2}\right) & 1\\
n=5 & \sin \left(\frac{25 \pi}{8} \pi\right)=\sin \left(\frac{ \pi}{8} \pi\right) &  \frac{\sqrt{2+\sqrt{2}}}{2} \\
n=6 & \operatorname{an}\left(\frac{30 \pi}{8}\right)=\sin\left(\frac{-\pi}{4}\right) & \frac{-1}{\sqrt{2}}\\
n=7 & \sin\left(\frac{35 \pi}{8}\right)=\sin\left(\frac{3 \pi}{8}\right) & \frac{\sqrt{2+\sqrt{2}}}{2}\\
n=8 & \sin\left(5 \pi\right) & 0\\
n=9 & \sin\left(\frac{45 \pi}{8}\right)=\sin\left(\frac{5 \pi}{8}\right) & \frac{\sqrt{2+\sqrt{2}}}{2}\\
n=10 & \sin\left(\frac{50 \pi}{8}\right)=\sin\left(\frac{\pi}{4}\right) & \frac{1}{\sqrt{2}}\\
n=11 & \sin \left(\frac{55 \pi}{8} \pi\right)=\sin \left(\frac{- \pi}{8}\right) & -\frac{\sqrt{2-\sqrt{2}}}{2}\\
\end{array}\\
...
\end{aligned}
$$
When I try to put it with coefficients I get:
$$
\begin{aligned}
7+\frac{80}{\pi} &\left(\frac{1}{1} \frac{\sqrt{2+\sqrt{2}}}{2}+\frac{1}{2} \frac{1}{\sqrt{2}}+\frac{1}{3} \frac{\sqrt{2-\sqrt{2}}}{2}-\frac{1}{4} \frac{\sqrt{2+\sqrt{2}}}{2}+\frac{1}{5} \frac{\sqrt{2+\sqrt{2}}}{2}\right.\\
&+\frac{1}{6} \frac{1}{\sqrt{2}}+\frac{1}{7} \frac{\sqrt{2+\sqrt{2}}}{2}-\frac{1}{8} \cdot 0+\frac{1}{9} \frac{\sqrt{2+\sqrt{2}}}{2}-\frac{1}{10} \frac{1}{\sqrt{2}}-\frac{1}{11} \frac{\sqrt{2-\sqrt{2}}}{2} ...)
\end{aligned}
$$
I am sorry in advance for typos, I have a tendency to miss them when typing latex!
Question: How do I see my series for f(5)? I can't identify any Taylor series or familiar sum. Are there particular techniques to distinguish a series?
 [/INST] The answer was clearly stated by @WA Don in my previous question and I wish I can select it as a solution for this question as well.
I did not understood the condition (2.) at he time, because I simply did not understood how one could take all Fourier coefficients (I still don't actually!).
From WA Don answer:
2.  If $f$ is continuous and the sum of the Fourier coefficients converges absolutely (i.e. $\sum |a_n| + |b_n|$ converges) then the  Fourier series converges to $f(x)$.
The same theorem was available in my book, Differential Equations with Boundary-Value Problems:

In conclusion, since the function $g(x) = 5(x) + 7$ on the interval $[-8,8]$ is continuous, its Fourier series $ 7 + \sum_{n=1}^\infty \frac{(-1)^{n+1}80}{n\pi} \sin \left( \tfrac{1}{4}n\pi \right)$ converges to $5(x) + 7$, that is f(5) = 32.
</s>