Exercise

1. Find the radius of convergence of the following power series.
$$\begin{array}{l} (a) \ \sum \frac{nx^{n}}{3^{n}} \ \ (b) \ \sum \frac{n^{n}x^{n}}{n!} \end{array}$$
2. Show the following.
$$\begin{array}{l} (a) \ \sum_{n=0}^{\infty}x^{n} = \frac{1}{1 ... ...}\frac{(-1)^{n}x^{n+1}}{n+1}, \ \vert x\vert \leq 1 \end{array}$$
3. Suppose that $f(x) = \sum_{n=1}^{\infty}\frac{\sin{nx}}{n^3}$. Then show the following.

(a) $f(x)$ is uniformly convergent on $-\infty < x < \infty$.

(b) $f^{\prime}(x) = \sum_{n=1}^{\infty}\frac{\cos{nx}}{n^2}$

(c) $\int_{0}^{\pi}f(x)dx = \sum_{n=1}^{\infty}\frac{2}{(2n-1)^{4}}$