Let $\displaystyle g_k(x)= \left\{\begin{array}{cc} \frac {1}{k^2},&\mbox{ if }

|x| \leq k\\ \frac {1}{x^2}, & \mbox{ if } |x|>k\end{array}\right. $

Does the series $\displaystyle \sum ^ \infty _{k=1} g_k(x) $ converges pointwise or uniformly?

Proof so far.

Define the partial sum $\displaystyle s_n= \sum ^n_{k=1} g_k (x) $.

Now, in the case that $\displaystyle |x| \leq k $, we have $\displaystyle s_n(x)= \sum ^n_{k=1} \frac {1}{k^2} = \sum ^n_{k=1} ( \frac {1}{k})^2$

Now, does this one converges to $\displaystyle \frac {1}{k^2} $, if it does then it is pointwise.

In the case that $\displaystyle |x| > k $, then $\displaystyle s_n = \sum ^ n_{k=1} \frac {1}{x^2} = \frac {n-1}{x^2} = \frac {n}{x^2} - \frac {1}{x^2}$

Well, then this guy doesn't converge to $\displaystyle g_k$, so it ain't pointwise convergence then?

Thanks, people, I'm really lost in series convergence here...