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Thread: Does this series converge?

  1. #1
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    Does this series converge?

    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...
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    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?
    if $\displaystyle |x| > k,$ then $\displaystyle g_k(x)=\frac{1}{x^2} < \frac{1}{k^2}.$ thus: $\displaystyle g_k(x) \leq \frac{1}{k^2},$ for all $\displaystyle x \in \mathbb{R}, \ k \in \mathbb{N}.$ hence by Weierstrass M-test your series is uniformly convergent. Weierstrass M-test
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