By expanding $\left(t^{2}+4 \pi n^{2}\right)^{-1}$ and interchanging sums (justifying this, if you can, just interchanging, if not) deduce that
$$
2\left(1-e^{-t}\right)^{-1}=1+2 t^{-1}+\sum_{m=0}^{\infty} c_{m} t^{m}
$$
where $c_{2 m}=0$ and
$$
c_{2 m+1}=a_{2 m+1} \sum_{n=1}^{\infty} n^{-2 m}
$$
for some value of $a_{2 m+1}$ to be given explicitly.
Show that the nth Fourier sum of $F$
$$
S_{n}(F, t)=\sum_{r=-n}^{n} \hat{F}(r) \exp (i r t)=2 \sum_{r=1}^{n} \frac{1}{r} \sin r t
$$
Explain why
$$
S_{n}(F, \tau / n)=2 \frac{\pi}{n} \sum_{r=1}^{n} \frac{1}{\frac{\pi}{n}} \sin \frac{r \pi}{n} \rightarrow \int_{0}^{\tau} \frac{\sin x}{x} d x
$$
as $n \rightarrow \infty$.
Sketch the behaviour of the function
$$
G(\tau)=\int_{0}^{\pi} \frac{\sin x}{x} d x
$$
MATH4069 COURSE NOTES ：
(i) If $\varepsilon>0$, show that
$$
\int_{-\infty}^{\infty} \frac{\sin \lambda x}{x} d x \rightarrow \int_{-\infty}^{\infty} \frac{\sin x}{x} d x
$$
as $\lambda \rightarrow \infty$.
(ii) If $\pi \geq \epsilon>0$, show, by using the estimates from the alternating senies test, or otherwise, that
$$
\int_{-e}^{e} \frac{\sin \left(\left(n+\frac{1}{2}\right) x\right)}{\sin \frac{x}{2}} d x \rightarrow \int_{-\pi}^{\pi} \frac{\sin \left(\left(n+\frac{1}{2}\right) x\right)}{\sin \frac{x}{2}} d x=2 \pi
$$
as $n \rightarrow \infty$.
(iii) Show that
$$
\left|\frac{2}{x}-\frac{1}{\sin \frac{1}{2} x}\right| \rightarrow 0
$$
as $x \rightarrow 0$. and dedoce that
$$
\int_{0}^{\infty} \frac{\sin x}{x} d x=\frac{\pi}{2}
$$