Answer to Question #310393 in Real Analysis for Dhruv bartwal

Question

Test whether the series ∞Σn=0 1/(n^5+x^3) converges uniformly or not

Answer 1

The series converges for all x such that $n^5+x^3\ne 0$ for $n\geq 1$ .

Consider

$\left| S_n\left( x \right) -S_{n-1}\left( x \right) \right|=\left| \frac{1}{n^5+x^3} \right|$

For $x=-\left( n^5-1 \right) ^{1/3}$

$\left| S_n\left( x \right) -S_{n-1}\left( x \right) \right|=1$

The series doesn’t converge uniformly by the Cauchy criterion.