I am currently investigating the convergence of the following function,

$f(x)=\sum\limits_{k=1}^{\infty} \dfrac{x^{k}+\sin(k)}{k^{2}}$

for different "senses". I have shown that $f(x)$ converges uniformly and pointwisely for its domain. All I have left to do is investigate whether it converges in the $L^{p}$ sense. The problem is, I have absolutely no idea what this means! My knowledge on this $L^{p}$ is very minimal and I can not seem to find anything on the www or Rudin. If someone could explain to me what this $L^{p}$ business is it would be greatly appreciated. A few sub-questions I have include:

  • How does it differ from uniform convergence?
  • Any geometrical interpretations?
  • I know that $p\geq1%$. How does every each value of $p$ affect convergence?
  • Are there any relationships between each $L^{p}$?

If anyone knows of some nice references to my specific questions, that also would be greatly appreciated.

Thank you in advanced to all for your time.

  • 1
    $\begingroup$ do you know what the $L^p$ norm is $\|f\|_p=(\int|f|^pdx)^{\frac{1}{p}}$ $\endgroup$ – Ellya Apr 9 '14 at 15:25
  • $\begingroup$ Ok, thanks @ellya. That is a start to my understanding. Anymore help? $\endgroup$ – Gustavo Louis G. Montańo Apr 9 '14 at 21:52
  • $\begingroup$ what does f(x) converge to? $\endgroup$ – Ellya Apr 9 '14 at 21:55
  • $\begingroup$ @ellya. It converges, but I do not to what to. The first term, $\frac{x^{k}}{k{2}}$ converges via the ratio test and letting |x| < 1, and the second term converges via the W-M Test. Are you asking for the function that it converges to? I do not know that. Can you help me find it, perhaps? I have proved Uniform and PW convergence via definition of delta-epsilon $\endgroup$ – Gustavo Louis G. Montańo Apr 9 '14 at 22:03

Well we can see that $|f(x)|=|\sum_{k=1}^\infty\frac{x^k+sin(k)}{k^2}|\leq\sum_{k=1}^\infty|\frac{x^k+sin(k)}{k^2}|\leq \sum_{k=1}^\infty|\frac{x^k}{k^2}|+\sum_{k=1}^\infty\frac{1}{k^2}=\sum_{k=1}^\infty|\frac{x^k}{k^2}|+\frac{\pi^2}{6}$

Now like you said if we restrict to $x\in[0,\frac{1}{2})$ (You'll see why later, then we have $|f(x)|\leq\sum_{k=1}^\infty|\frac{x^k}{k^2}|+\frac{\pi^2}{6}\leq \sum_{k=1}^\infty x^k+\frac{\pi^2}{6}=\frac{1}{1-x}+\frac{\pi^2}{6}$

Now as I said $\|f\|_p=(\int |f|^pdx)^{\frac{1}{p}}$.

Let us look at $L^1$, here $\|f\|_1=\int_0^\frac{1}{2}|f(x)|dx=\int_0^\frac{1}{2}\frac{1}{1-x}+\frac{\pi^2}{6}dx=(-\ln(1-x)+\frac{\pi^2}{6}x)|_0^1=-\ln{\frac{1}{2}}+\frac{\pi^2}{6}=\ln(2)+\frac{\pi^2}{6}\lt\infty$, so $f\in L^1$.

This is all i've thought up so far, but I hope it helps a bit.


I've thought of a much better way to do this.

We as before restrict to $|x|\le 1$, then:


Thus $\int_{-1}^{1}|f(x)|^pdx\leq\int_{-1}^{1}(\frac{\pi^2}{3})^pdx=2(\frac{\pi^2}{3})^p\lt\infty$, Thus $f\in L^p([-1,1])$

  • $\begingroup$ Hey @ellya, yes that has made more sense. A few questions. What you did in the first 4 lines, what did you achieve? Did you show that my original function converges to $\sum_{k=1}^\infty|\frac{x^k}{k^2}|+\frac{\pi^2}{6}$? If so, I can not see why. Sure, I know the function converges, but would not be satisfied to conclude that it converges to that. Now, the domain is [-1,1]. And I am curious to know why you only went from 0 -> (1/2).How about L2 and L3 and so on? Thanks ! $\endgroup$ – Gustavo Louis G. Montańo Apr 9 '14 at 23:04
  • $\begingroup$ Firstly, what I found was an upper bound for f, which allowed me to show that f converges under the $L^1$ norm. I took $(0,1/2)$ literally because it simplifies this integral, Im sure with more effort you can extend the interval. As for L^p in general, the integral just becomes a bit more complicated, as we take the pth power in the integral. $\endgroup$ – Ellya Apr 9 '14 at 23:11
  • $\begingroup$ I see. With the $L^{p}$ is it possible though? Because I am required to see if it converges in the $L^{p}$ sense. So, how can I come to a final conclusion? Any hint - I don't expect you to do it for me. $\endgroup$ – Gustavo Louis G. Montańo Apr 9 '14 at 23:18
  • $\begingroup$ Try what I did, firstly on (0,1/2) BUT do the integral of $|f|^p$ and see what happens,hopefully it will be finite, after that try making the interval larger. $\endgroup$ – Ellya Apr 9 '14 at 23:21
  • $\begingroup$ Also what was the domain of f meant to be? Its not in your post. $\endgroup$ – Ellya Apr 9 '14 at 23:24

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