Find the values of x for which the series $\sum_{n=1}^{\infty}\frac{x^{a}}{1+n^2x^2}, x>0, a\geq 0$ converges uniformly on (1)$[0,1]$ and (2)$[0, \infty)$
Let $f(x)=\frac{x^{a}}{1+n^2x^2} \\ f'(x) = \frac{ax^{a-1}(1+n^2x^2)-x^a2n^2x}{(1+n^2x^2)^2} = 0 \\x=c=\frac{\sqrt a}{\sqrt (2-a)}\frac{1}{n} \\f(c) = \left(\frac{a}{2-a}\right)^{\frac{a}{2}}\frac{1}{n^a}\frac{2-a}{a}$
Series is convergent for all $a \geq 1$ and divergent for $a<1$
Now how to see uniform convergence in given intervals. a is also coming into picture