11

There is an infinite sum given: $$\sum_{n=1}^{\infty}\frac{1}{n^22^n}$$ It should be solved using integration, derivation or both. I think using power series can help but I don't know how to finish the calculation. Any help will be appreciated!

jimjim
  • 9,675
Hendrra
  • 2,868
  • 1
  • 18
  • 35

3 Answers3

15

Note that we have $\int_0^x t^{n-1}\,dt=\frac{x^n}{n}$. Then, we can write

$$\begin{align} \sum_{n=1}^\infty \frac{x^{2n}}{n^2}&=\sum_{n=1}^\infty \int_0^x t^{n-1}\,dt\int_0^x s^{n-1}\,ds\\\\ &=\int_0^x\int_0^x \frac{1}{1-st}\,ds\,dt\\\\ &=-\int_0^{x} \frac{\log(1-sx)}{s}\,ds\\\\ &=-\int_0^{x^2} \frac{\log(1-s)}{s}\,ds\\\\ &=\text{Li}_2(x^2) \end{align}$$

Evaluating at $x=1/\sqrt{2}$ yields

$$\bbox[5px,border:2px solid #C0A000]{\sum_{n=1}^\infty \frac{1}{n^2\,2^n}=\text{Li}_2(1/2)=\frac{\pi^2}{12}-\frac12\log^2(2)}$$

And we are done!

To evaluate $\text{Li}_2(1/2)$, we exploit the relationship

$$\text{Li}_2(1-x)=-\text{Li}_2\left(1-\frac1x\right)-\frac12\log^2(x)$$

Letting $x=1/2$ yields

$$\text{Li}_2(1/2)=-\text{Li}_2\left(-1\right)-\frac12\log^2(1/2)=\frac{\pi^2}{12}-\frac12\log^2(2)$$

where we used

$$\begin{align} \text{Li}_2(-1)&=-\int_0^{-1}\frac{\log(1-x)}{x}\,dx\\\\ &=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^2}\\\\ &=\frac{\pi^2}{12} \end{align}$$

Mark Viola
  • 179,405
  • That's a nice solution! Thanks. However I must ask about $\sum_{n=1}^{\infty}\frac{x^{2n}}{n^2}$. Why are we considering such a sum? – Hendrra Apr 23 '17 at 19:43
  • You're welcome. My pleasure. The value of that sum when $x=1/\sqrt 2$ is $\sum_{n=1}^\infty \frac{1}{n^2,2^n}$. – Mark Viola Apr 23 '17 at 19:45
  • The value truly is $\sum_{n=1}^{\infty}\frac{1}{n^22^n}$! Thus I have the next question. Why did you decided to take an $x = \frac{1}{\sqrt{2}}$? Just because it works? – Hendrra Apr 23 '17 at 19:48
  • Well, yes. We took $x=1/\sqrt 2$ in order that we would have the sum of interest. – Mark Viola Apr 23 '17 at 19:51
  • Thank you. Your comments were really helpful. I have to ask about one more thing. How did you get $\int_0^x\int_0^x \frac{1}{1-st},ds,dt$? – Hendrra Apr 23 '17 at 19:54
  • You're welcome again. And good question. We sum the geometric series $$\sum_{n=1}^\infty (st)^{n-1}=\frac{1}{1-st}$$ – Mark Viola Apr 23 '17 at 19:56
  • Thanks! And the last question - how did you integrate $\int_0^x\int_0^x \frac{1}{1-st},ds,dt$? – Hendrra Apr 23 '17 at 20:11
  • Why does the upper limit of the integrand become squared in the last step of your initial derivation. – Tyberius Apr 23 '17 at 20:19
  • 1
    @Tyberius The first integral yields $$\int_0^x \int_0^x \frac{1}{1-st},dt,ds=-\int_0^x \frac{\log(1-sx)}{s},ds=-\int_0^{x^2}\frac{\log(1-s)}{s},ds=\text{Li}_2(x^2)$$ – Mark Viola Apr 23 '17 at 20:20
11

Another way to calculate it: consider $$ f(x)=\sum_{n=1}^{+\infty}\frac{x^n}{n^2}. $$ Differentiating w.r.t. $x$ gives $$ f'(x)=\sum_{n=1}^{+\infty}\frac{x^{n-1}}{n}=\frac{1}{x}\sum_{n=1}^{+\infty}\frac{x^n}{n}\quad\Rightarrow\quad xf'(x)=\sum_{n=1}^{+\infty}\frac{x^n}{n}\quad\Rightarrow\quad (xf'(x))'=\sum_{n=1}^{+\infty}x^{n-1}=\frac{1}{1-x}. $$ Now integrating with $f(0)=0$ $$ xf'(x)=-\ln(1-x)\quad\Rightarrow\quad f(x)=-\int_0^x\frac{\ln(1-t)}{t}\,dt. $$ Motivation for termwise differentiation for power series is straightforward.

A.Γ.
  • 29,518
2

My answer to a duplicate question says:

$$\frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}$$ Integrating from 0 to t we get $$\int_{0}^{t}\frac{1}{(1-x)}dx=\sum_{n=0}^{\infty}\int_{0}^{t} x^{n}dx$$$$-\ln(1-t)=\sum_{n=1}^{\infty} \frac{t^{n}}{n}$$ Dividing by t and integrating $$\int_{0}^{0.5}-\frac{\ln(1-t)}{t}dt=\sum_{n=1}^{\infty}\int_{0}^{0.5} \frac{t^{n-1}}{n}dt=\sum_{n=1}^{\infty} \frac{1}{n^{2}2^{n}}$$ This on calculating is $$ \dfrac{\pi^2}{12}-\dfrac{ln^2(2)}{2},$$

Mr.HiggsBoson
  • 1,766
  • This answer is identical to your answer to this question. If the questions are the same, this one should be flagged as a duplicate. If not, then it would be more efficient and less noisy to cite or quote (with attribution) from the other answer. The link between the answers would also serve as a link between two closely related questions. In this case, the questions were duplicates. – robjohn May 26 '19 at 01:31
  • @robjohn So should I change anything ??. I posted it here because the other one was marked as duplicate. Should I remove it from here?? – Mr.HiggsBoson May 26 '19 at 03:12
  • Even though the other question was closed, your answer there is still active and getting votes. One should be removed, but you might edit one to reference the other; you might just say something like, "my answer to a duplicate question says..." – robjohn May 26 '19 at 05:28
  • Okay , I will link my answers – Mr.HiggsBoson May 26 '19 at 09:55