My Math Forum Another series question-- on uniform convergence.

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 December 30th, 2011, 08:22 PM #1 Senior Member   Joined: Nov 2011 Posts: 100 Thanks: 0 Another series question-- on uniform convergence. Claim: The series $\sum_{n=0}^\infty \frac{x^2}{(1+x^2)^n}$ converges uniformly on $[a,\infty)$ for every $a>0$, but not uniformly on $[0,b]$ for every $b>0$. I wanted to try the Weierstrauss M-test, but the only bound I could find for the functions is $\frac{x^2}{(1+x^2)^n}\leq \frac{1}{\sqrt{n-1}}$, but $\sum\frac{1}{\sqrt{n-1}}$ does not converge (compare to harmonic series). The other thing I was trying was showing that the partial sums somehow satisfy the Cauchy criterion-- but again, I wasn't getting anywhere. Any hints or ideas? Thanks!
 December 31st, 2011, 02:02 PM #2 Global Moderator   Joined: May 2007 Posts: 6,378 Thanks: 542 Re: Another series question-- on uniform convergence. You have a geometric series with r=1/(1+x^2). As long as x is bounded away from zero, the convergence will be uniform, since convergence at the lower bound is slowest. For the interval with lower end 0, the problem lies in the fact that the convergence is slower and slower toward that end.

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