# How to calculate the value of $\sum\limits_{k=0}^{\infty}\frac{1}{(3k+1)\cdot(3k+2)\cdot(3k+3)}$?

How do I calculate the value of the series $$\sum_{k=0}^{\infty}\frac{1}{(3k+1)\cdot(3k+2)\cdot(3k+3)}= \frac{1}{1\cdot2\cdot3}+\frac{1}{4\cdot5\cdot6}+\frac{1}{7\cdot8\cdot9}+\cdots?$$

• The answer is $\frac{\pi\sqrt3}{12}-\frac{\ln3}{4}.$ – Michael Rozenberg Feb 10 '18 at 18:38
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By making use of the integral $$\int_{0}^{1} \frac{(1-x)^2}{1-x^3} \, dx = \frac{1}{2} \, \left(\frac{\pi}{\sqrt{3}} - \ln 3 \right)$$ one can take the following path. \begin{align} S &= \sum_{k=0}^{\infty} \frac{1}{(3k+1)(3k+2)(3k+3)} \\ &= \sum_{k=0}^{\infty} \frac{\Gamma(3k+1)}{\Gamma(3k+4)} = \frac{1}{2} \, \sum_{k=0}^{\infty} B(3, 3k+1), \end{align} where $B(n,m)$ is the Beta function, which leads to \begin{align} S &= \frac{1}{2} \, \sum_{k=0}^{\infty} \, \int_{0}^{1} t^{2} \, (1-t)^{3k} \, dt \\ &= \frac{1}{2} \, \int_{0}^{1} \frac{t^{2} \, dt}{1- (1-t)^{3}} \\ &= \frac{1}{2} \, \int_{0}^{1} \frac{(1-x)^{2} \, dx}{1- x^3} \hspace{15mm} x = 1 - t \\ &= \frac{1}{4} \, \left(\frac{\pi}{\sqrt{3}} - \ln 3 \right). \end{align}

• How do we justify exchanging the limit of the series and the integral here? AFAIK, the convergence of the series isn't uniform on [0,1]. – Anu Feb 10 '18 at 20:07

The answer is $\frac{\pi\sqrt3}{12}-\frac{\ln3}{4}.$

See the similar problem (problem 2) here: http://www.imc-math.org.uk/imc2010/imc2010-day1-solutions.pdf

You can calculate the partial sum using $$\frac{1}{(3k+1)(3k+2)(3k+3)}=\frac{1}{2}\frac{1}{3k+1}-\frac{1}{3k+2}+\frac{1}{2}\frac{1}{3k+3}$$

• Where does this come from? – Jack Moody Feb 10 '18 at 18:27
• Partial decomposition. For example the coef before $1/(3k+1)$ is $$a=\frac{1}{(-1+2)(-1+3)}=\frac{1}{2}$$. Hope i did not make mistakes for the rest. – Atmos Feb 10 '18 at 18:29
• Ah, I see. Thanks for the clarification. – Jack Moody Feb 10 '18 at 18:30
• you can't use this directly: all individual terms are divergent. – dezdichado Feb 10 '18 at 18:56
• @Atmos and? There is not any cancellation and the partial sum does not have any closed form. I would be surprised if you can calculate it, just using the decomposition. – dezdichado Feb 10 '18 at 19:05


Moreover, $\ds{H_{-1/3} = H_{2/3} - 3/2\ \pars{~recurrence~}}$.

$\ds{H_{-1/3} = \overbrace{\braces{3\bracks{1 - \ln\pars{3}}/2 + \root{3}\pi/6}}^{\ds{H_{2/3}}}\ -\ 3/2 = -3\ln\pars{3}/2 + \root{3}\pi/6}$. The $\ds{H_{2/3}}$ value is given in a table. Otherwise, it can be evaluated by means of the Gauss Digamma Theorem.

Finally, \begin{align} &\bbox[10px,#ffd]{\ds{% \sum_{k = 0}^{\infty}{1 \over \pars{3k + 1}\pars{3k + 2}\pars{3k + 3}}}} = {\root{3} \over 18}\,\pi + {1 \over 6}\bracks{- {3\ln\pars{3} \over 2} + {\root{3} \over 6}\,\pi} \\[5mm] = &\ \bbx{\root{3}\pi - 3\ln\pars{3} \over 12} \approx 0.1788 \end{align}