Results 1 to 2 of 2

Math Help - Does this series converge?

  1. #1
    Super Member
    Joined
    Mar 2006
    Posts
    705
    Thanks
    2

    Does this series converge?

    Let  g_k(x)= \left\{\begin{array}{cc} \frac {1}{k^2},&\mbox{ if }<br />
|x| \leq k\\ \frac {1}{x^2}, & \mbox{ if } |x|>k\end{array}\right.

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

    Proof so far.

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

    Now, in the case that |x| \leq k , we have 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  \frac {1}{k^2} , if it does then it is pointwise.

    In the case that  |x| > k , then  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 g_k, so it ain't pointwise convergence then?

    Thanks, people, I'm really lost in series convergence here...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by tttcomrader View Post
    Let  g_k(x)= \left\{\begin{array}{cc} \frac {1}{k^2},&\mbox{ if }<br />
|x| \leq k\\ \frac {1}{x^2}, & \mbox{ if } |x|>k\end{array}\right.

    Does the series  \sum ^ \infty _{k=1} g_k(x) converges pointwise or uniformly?
    if |x| > k, then g_k(x)=\frac{1}{x^2} < \frac{1}{k^2}. thus: g_k(x) \leq \frac{1}{k^2}, for all x \in \mathbb{R}, \ k \in \mathbb{N}. hence by Weierstrass M-test your series is uniformly convergent. Weierstrass M-test
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 4
    Last Post: June 26th 2010, 07:04 AM
  2. Does this series converge?
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 5th 2008, 01:10 AM
  3. Does the series converge?
    Posted in the Calculus Forum
    Replies: 4
    Last Post: November 4th 2008, 01:22 PM
  4. When does the series converge
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 3rd 2008, 08:21 PM
  5. Series Converge Value
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 2nd 2008, 04:58 PM

Search Tags


/mathhelpforum @mathhelpforum