Use the Weierstrass M-test to show that the given series of functions converges uniformly on the given set:

Question

Use the Weierstrass M-test to show that the given series of functions converges uniformly on the given set:

(a) $\sum_{k=1}^\infty \frac{x^k}{k(k+1)}$ on $[-1,1]$

(b) $\sum_{k=0}^\infty (\frac{x+2}{5})^k$ on $[-2,2]$

(c) $\sum_{k=0}^\infty \frac {(x-2)^k}{3^k}$ on $[0,4]$

Answer 1

$$ (a) \left|\frac{x^k}{k(k+1)}\right| \le \frac{1}{k(k+1)},\;\; x\in [-1,1]. $$ $$ (b) \left|\left(\frac{x+2}{5}\right)^k\right| \le \left(\frac{4}{5}\right)^k,\;\; x\in[-2,2] $$ $$ (c) \left|\frac{(x-2)^k}{3^k}\right| \le \left(\frac{2}{3}\right)^k,\;\ x\in [0,4] $$