Series of Functions

(1)

(a) Show that $\displaystyle g(x)= \sum^{\infty}_{n=1}{cos\frac{(2^nx)}{2^n}}$ is continuous on all of $\displaystyle \mathbb{R}$

(b) Prove that $\displaystyle h(x)= \sum^{\infty}_{n=1}{\frac{x^n}{n^2}}$ is continuous on $\displaystyle [-1,1]$.

(2) Observe that the series $\displaystyle f(x)=\sum^{\infty}_{n=1}{\frac{x^n}{n}}=x+\frac{x^ 2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+...$ converges for every $\displaystyle x$ in the half-open interval $\displaystyle [0,1)$ but does not converge when $\displaystyle x=1$. For a fixed $\displaystyle x_0 \in (0,1)$, explain how we can still use the Weierstrass M-Test to prove that $\displaystyle f$ is continuous at $\displaystyle x_0$.

(3) Let $\displaystyle h(x)=\sum^{\infty}_{n=1}{\frac{1}{x^2+n^2}}$

Show that $\displaystyle h$ is a continuous function defined on all of $\displaystyle \mathbb{R}$.

Thanks!

You prove these problems uses the Weierstrass test. For example, in 1(a) we see that $\displaystyle \left| \tfrac{\cos 2^n x}{2^n} \right| \leq \tfrac{1}{2^n}$ and the series of $\displaystyle \tfrac{1}{2^n}$ converges. Thus, this series of fuctions converges uniformly. Since each function is continous we see that the limit (the infinite sum) is continous because the uniform limit of continous functions is continous.