Derivative of a power series

Question

Suppose the sequence $(b_k) , k\geq 0$ satisfies $\sum k|b_k| < \infty$, then show that $\sum_{k=0}^\infty b_kx^k$ converges uniformly to a function $g$ on $|x| \leq 1$ and that $g'(x) = \sum_{k=1}^\infty kb_kx^{k-1} $ for all x, $|x| \leq 1$.

Answer 1

Apply Weierstrass M-Test to get absolute and uniform convergence for $\;|x|<1\;$:

$$\sum_{k=0}^\infty\left|b_kx^k\right|\le\sum_{k=0}^\infty|b_k|\le\sum_{k=0}^\infty k|b_k|\stackrel{\text{given}}<\infty$$

Now use that a uniform convergent (power) series can be differentiated term-by-term...