Uniform convergence of power series $\sum_{n=1}^\infty \frac {x^n}{(n+1)(n+2)}$

Question

Prove the uniform convergence of power series $$\sum_{n=1}^\infty \frac {x^n}{(n+1)(n+2)}$$ on the closed interval $[-1,1]$.

The radius of convergence $$R = \lim_{n\to\infty} | \frac{a_n}{a_{n+1}}| = \lim_{n\to\infty} |\frac {(n+2)(n+3)}{(n+1)(n+2)}| = 1$$
But how do you prove it is uniformly convergent on $[-1,1]$ ?

Answer 1

You need to show that the series also converge for $x = \pm 1$, since radius of convergence is inconclusive at these points.

To show uniform convergence, observe that $$ \left|\frac {x^n}{(n+1)(n+2)}\right| \leq \frac{1}{(n+1)(n+2)} $$ and $\sum_{n=1}^\infty \frac{1}{(n+1)(n+2)}$ converges. Applying the Weierstrass M test, you can establish the uniform convergence of $\frac {x^n}{(n+1)(n+2)}$.

Answer 2

Hint. From the inequality $$ \left|\sum_{n=1}^\infty \frac {x^n}{(n+1)(n+2)}\right|\leq \sum_{n=1}^\infty \frac 1{(n+1)^2}<\infty $$ one deduces the uniform convergence of the series on $[-1,1]$.