Question

Use Weierstrass’ M-Test to prove that the series

Infinity

∑ n^2x^n

n=1

converges uniformly in the interval [0,1/5]

Answer 1

We have

$\left| n^2x^n \right|\leqslant \frac{n^2}{5^n}$

The series $\sum{\frac{n^2}{5^n}}$ converges by D’Alembert’s test since

$\underset{n\rightarrow \infty}{\lim}\frac{\frac{n^2}{5^n}}{\frac{\left( n+1 \right) ^2}{5^{n+1}}}=5>1$

Then by Weierstrass’ M-Test the series $\sum{n^2x^n}$ converges uniformly.