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Question: Consider the series $g_n(x)=\sum_{k=0}^n\cfrac{x^2}{(1+x^2)^k}$ Prove that the series converges pointwise to the function $$g(x)=\begin{cases} 0 & \text{ if } x=0 \\ 1+x^2 & \text{ if } x \neq 0 \end{cases}$$ but the convergence is not uniform on any interval containing $0$ on its interior.

Help? Not even sure what the first step is on this one.

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    You might first calculate the sum. Let $x\ne 0$. Then our series is a convergent geometric series. To check whether the convergence is uniform, look at the remainder if you stop summing at $m$. You will find a heavy dependence on $x$. Or else you can use *properties* of functions that are limits of uniformly convergent series.2012-12-08

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