$$ Exercise 1 (2 pt) For $f \in \mathcal{R}[0, M], M>0$, recall that the improper integral $\int _{0}^{+\infty} f(x) d x$ converges if and only if $$ \lim _{M ightarrow+\infty} \int_{0}^{M} f(x) d x exists. Consider the function $f(x) \in \mathcal{C}[0,+\infty $ defined by $$ f(x)=\left\{\begin{array}{11} \frac{\sin x}{x}, & x>0 W 1, & x=0 \end{array} ight. $$ 1) Show that $\int_{0}^{+\infty} f(x) d x$ converges if and only if $\left(S_{n} ight)$ converges, where $$ S_{n}=\int_{0}^{n \pi} \frac{\sin x}{x} d x $$ Hint: $/\sin x / x \leq|1 / x[$. 2) Write $5_{n}=\sum_{i=1}^{n}(-1)^{i-1} a_{i}$ where $$ a_{i}=(-1)^{i-1} \int_{(i-1) \pi}^{i \pi} \frac{\sin x}{x} d x. $$ Show that $a_{i} \geq 0$ and $\left(a_{n} ight) $ is decreasing. 3) Conclude that $\int_{0}^{+\infty} f(x) d x$ converges. CS.JG.061