Question

How to show that series summation[1/(1+xn^2)] does not converge uniformly on (0,infinity) using Cauchy criterion?

Answer 1

\[\sum_{n=0}^{\infty}1/(1+n ^{2}x)\]

Answer 2

\[ \left | \frac 1{1 + (n+1)^2x} + \frac 1{1 + (n+2)^2x} + .... +\frac 1{1 + (n+p)^2x} \right | < \\
\left | \frac 1{xn^2} + \frac 1{xn^2}+ ... + \frac 1{xn^2} \right | = |\frac p{xn^2}| < \epsilon \text{ for some n >= N}\]