Proving uniform convergence of a simple series $\sum_{k=1}^{\infty} k^2 x^k$

Question

I have a math question that I desperately need help on. I need to prove that a series is uniform convergent on $(-1,1)$. The series is $$\sum_{k=1}^{\infty} k^2 x^k$$ I tried to use the $M$-test for series convergence but failed. Are there other methods of proving or the definition needs to be used in this case. Thanks for answering.

Answer 1

Hint: Try applying the ratio test.

Answer 2

If the series $$\sum_{k\ge1}f_k(x)$$ is uniformly convergent on an interval $I$ then the sequence $(f_k)_{k\ge1}$ is uniformly convergent to $0$.

In your case the sequence $(k^2 x^k)$ isn't uniformly convergent to $0$ since

$$\sup_{x\in(-1,1)}|k^2x^k|=k^2\xrightarrow{k\to\infty}\infty$$ so the series isn't uniformly convergent on $(-1,1)$.

Answer 3

You can use either the ratio test or the root test. They both work equally well.

Ratio test: $a_n=n^2$, hence $\frac{a_n}{a_{n+1}}=\frac{n^2}{(n+1)^2}\to 1$, and hence the radius of convergence of the series is equal to $1$. This means that the power series converges for every $x\in (-1,1)$ and converges absolutely and uniformly in $[-r,r]$, for every $0<r<1$.