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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?

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    $\begingroup$ How can $\;x\;$ equal the running index? $\endgroup$
    – DonAntonio
    Jan 8, 2017 at 10:52

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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 $\varnothing$.

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