# Largest set where the series converges pointwisely

Find the largest set where the following series converges pointwisely? $$\sum_{n=1}^{\infty} \frac{1}{1+nx^4} .$$ Here what I did;

if $x=0 ,$ then obviously series diverges to $\infty$,

if $x=1,$ then series diverges since $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges.

I found this series converges only if $x=n$. Is this statement true? If not, how can I find the largest set ($=D$) in which the series convergent?

• How can $\;x\;$ equal the running index? Jan 8, 2017 at 10:52

If $|x| \le 1$, then $1+nx^4 \le 1 + n$, so the series diverges by comparison with the harmonic series.
If $|x| \ge 1$, then $1 + nx^4 \le x^4 + nx^4 = x^4(1+n)$, and so the series diverges (again by comparison with the harmonic series).
So the largest set is $\varnothing$.