For what values of $x$ does $\sum_{n=0}^\infty n^7 x^n$ converge?

For what values of $x$ does $\sum_{n=0}^\infty n^7 x^n$ converge?

I have attempted to solve this problem but I am not sure of my solution - I'd be glad if you could review it and give me some sort of feedback.

Consider the series $$\sum_{n=0}^\infty |n^7x^n| = \sum_{n=0}^\infty|n^7||x|^n$$ This series converges absolutely for $-1<x<1$, and so the initial series converges for such $x$ as well.

Now, I check $-1$ and $1$
If $x=1$, then this series clearly diverges. When $x=-1$, then it does not converge (All of the tests that I know are inconclusive - my claim is based on my observations. Any hints how to prove this?)

Now it's time for $x < -1$ and $x>1$. In both cases. $n^7x^n$ does not approach zero, and so the series does not converge.

And so the series converges for $x \in (-1,1)$

At $-1$ again the terms do not go to zero so it cannot converge. Otherwise your solution looks right.

Depending on how you show absolute convergence, you won't need to check $x < -1$ and $x > 1$. You can apply the ratio test to show that the radius of convergence must be $1$. Since the center of this series is $0$, the interval of convergence is "at least" $(-1, 1)$ and "at most" $[-1,1]$. So now all you need to do is check $x = \pm 1$.

For $x = -1$, the $n$th term test for divergence is applicable. Recall that if $\displaystyle \lim_{n\to+\infty} a_n \ne 0$, then $\displaystyle \sum_{n=1}^{+\infty} a_n$ does not converge. Actually, this is what you used for the case $x < -1$ and $x > 1$.

• In general the root test is more reliable than the ratio test, for example it easily gives you the correct behavior for $\sum_{n=0}^\infty x^{2^n}$. – Ian Nov 21 '17 at 19:41

One more test worth knowing.

The Alternating Series Test:

If all $a_n>0$ (or all $a_n <0$)

$S = \sum_\limits{n=0}^{\infty} (-1)^n a_n$ converges

If $a_n$ is monotonically decreasing (increasing if all $a_n$ are negative)

and $\lim_\limits{n \to \infty} a_n = 0$

Furthermore the partial sum

$S_k = \sum_\limits{n=0}^{k} (-1)^n a_n$ is approximates the (infinte) series with error bound $|S_k - S| < |a_{k+1}|$

For that matter, for any series $\sum a_n$ to converge

$\lim_\limits{n\to \infty} a_n = 0$ is a necessary condition.