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How could I find the sum of the series

$$ \sum_{n=1}^{\infty} \int_0^{\infty} \frac{\mathrm dx}{n(1+x^3)^n}$$

With: $$ \int_0^{\infty} \frac{\mathrm dx}{n(1+x^3)^n}=\frac{2\pi\Gamma(n-1/3)}{\Gamma(2/3)3^{3/2}n!}$$

(Previous post)

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    Hint: interchange sum and integral.2012-06-03

2 Answers 2

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Since integrals are taken over positive measurable functions we can interchange integration and summation $$ \sum_{n=1}^{\infty} \int_0^{\infty} \frac{dx}{n(1+x^3)^n}= \int_0^{\infty} \sum_{n=1}^{\infty}\left(\frac{1}{n(1+x^3)^n}\right)dx $$ Consider the following Taylor expansion $$ \log(1-q)=-\sum\limits_{n=1}^\infty\frac{q^n}{n}\qquad-1$q=(1+x^3)^{-1}$ we get $$ \sum\limits_{n=1}^\infty\frac{1}{n(1+x^3)^n}= -\log\left(1-\frac{1}{1+x^3}\right)= \log\left(\frac{1+x^3}{x^3}\right) $$ Hence $$ \sum_{n=1}^{\infty} \int_0^{\infty} \frac{dx}{n(1+x^3)^n}= \int_0^{\infty}\log\left(1+\frac{1}{x^3}\right)dx $$ Let's proceed to calculation of the last integral. For the first we use integration by parts $$ \int_0^{\infty}\log\left(1+\frac{1}{x^3}\right)dx= x\log\left(1+\frac{1}{x^3}\right)\biggr|_0^\infty- \int_0^{\infty}x\frac{d}{dx}\log\left(1+\frac{1}{x^3}\right)dx= 3\int_0^{\infty}\frac{1}{1+x^3}dx $$ Now lets make substitution $u=x^{-1}$, then we get $$ \int_0^{\infty}\frac{1}{1+x^3}dx=\int_0^{\infty}\frac{u}{1+u^3}dx $$ Hence, $$ \int_0^{\infty}\frac{1}{1+x^3}dx= \frac{1}{2}\left(\int_0^{\infty}\frac{1}{1+x^3}dx+\int_0^{\infty}\frac{x}{1+x^3}dx\right)= \frac{1}{2}\int_0^{\infty}\frac{1}{1-x+x^2}dx= $$ $$ \frac{1}{2}\int_0^{\infty}\frac{1}{\left(x-\frac{1}{2}\right)^2+\frac{3}{4}}d x= \frac{1}{2}\frac{2}{\sqrt{3}}\arctan\left(\frac{2x-1}{\sqrt{3}}\right)\Biggl|_0^\infty=\frac{2\pi}{3\sqrt{3}} $$ Finally, $$ \int_0^{\infty}\log\left(1+\frac{1}{x^3}\right)dx=3\int_0^{\infty}\frac{1}{1+x^3}dx=\frac{2\pi}{\sqrt{3}} $$

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    is it always ok to interchange sum and integral without knowing the nature of the integrant ?2012-06-03
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    @Ananda Well, not always. But if the series and its derivative is uniformly convergent, then we're cool.2012-06-03
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    Thank you very much for your clear answers!2012-06-04
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    @Chon, not at all.2012-06-04
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    Remark: Wolfram gives $$ \int_0^{\infty} \ln(1+1/x^3) \mathrm dx=2\pi/\sqrt{3}$$2012-06-04
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    Chon, thanks! I found mistake, see edits to the answer2012-06-04
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    The integral both answers put so much effort into evaluating is the case n=1 in the information given in the OP, and was derived in the linked thread.2012-06-06
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    $$ f(x) = \sum_{n=1}^{\infty} \frac{x^n}{n} \Rightarrow f'(x) = \sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x} $$ $$ f(x) = \int \frac{1}{1-x} \ dx = -\ln(1-x) + c $$ $$ but : f(0) = 0 \Rightarrow 0 + c = 0 \Rightarrow c = 0 \Rightarrow f(x) = \sum_{n=1}^{\infty} \frac{x^n}{n} = -\ln(1-x) $$2013-08-13
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    I'm old and forgetful, so I have a question: when you use your clever trick $\log(1+q)=\sum\dots$, setting $q=(1+x^{3})^{-1}$, don't you need to be careful with the limits of integration and take $\int^{\infty}_{\epsilon}\dots\,\mathrm{d}x$, then take the limit as $\epsilon\to0^{+}$? [I know this may be pedantic, but as I said -- I am old and forgetful, and I forgot whether you can skip it or if it's really necessary!]2013-08-13
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    @AlexNelson There is no need in this because I consider my integrals as Lebesgue integrals. Hence I can apply dominated convergence theorem in the first line. The second equality holds if $x\neq 0$. But since I'm talking about Lebesge integral I can forget about this points (becase it is of measure zero)2013-08-13
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    @Norbert, you are too clever for my feeble old mind! ;) Thanks for the explanation, I appreciate it greatly.2013-08-14
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Let us compute $I=\displaystyle\int_0^{+\infty}\log\left(\frac{1+x^3}{x^3}\right)\mathrm dx$.

  • The integration by parts with the functions $u(x)=\displaystyle\log\left(\frac{1+x^3}{x^3}\right)$ and $v'(x)=1$ yields $u'(x)=\displaystyle\frac{-3}{x(1+x^3)}$ and $v(x)=x$ hence $I=3\displaystyle\int_0^{+\infty}\frac{\mathrm dx}{1+x^3}$.

  • The change of variables $t=\displaystyle\frac1x$ yields $I=3\displaystyle\int_0^{+\infty}\frac{t\mathrm dt}{1+t^3}$.

  • Summing these yields $I=\displaystyle\frac32\int_0^{+\infty}\frac{(1+x)\mathrm dx}{1+x^3}=\frac32\int_0^{+\infty}\frac{\mathrm dx}{x^2-x+1}$.

  • The change of variables $2x-1=\sqrt3z$ yields $I=\displaystyle\frac32\cdot\frac2{\sqrt3}\int_{-1/\sqrt3}^{+\infty}\frac{\mathrm dz}{z^2+1}$, that is, $I=\sqrt3\cdot\left[\arctan z\right]_{-1/\sqrt3}^{+\infty}=\displaystyle\sqrt3\left(\frac\pi2+\frac\pi6\right)=\frac{2\pi}{\sqrt3}$.