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Question Number 147877 by Tawa11 last updated on 24/Jul/21

$$\mathrm{Find}\mathrm{the}\mathrm{sum}\mathrm{to}\mathrm{close}\mathrm{form}.$$$$\frac{1}{1.2.3}+\frac{1}{4.5.6}+\frac{1}{7.8.9}+...$$

Commented by Tinku Tara last updated on 24/Jul/21

$$\mathrm{Welcome}\mathrm{back}.\mathrm{Tawa}.$$🙂🙂

Commented by Tawa11 last updated on 24/Jul/21

$$\mathrm{Thank}\mathrm{you}\mathrm{sir}.\left(\stackrel{⌢..⌢}{⌣}\right)$$

Answered by Olaf_Thorendsen last updated on 24/Jul/21

$$\mathrm{S}=\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{1}{(3n+1)(3n+2)(3n+3)}$$$$\mathrm{S}=\frac{1}{8}\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{3}{(3n+1)(3n+2)}-\frac{5}{(3n+2)(3n+3)}+\frac{2}{(3n+3)(3n+1)}$$$$\mathrm{S}=\frac{1}{24}\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{1}{(n+\frac{1}{3})(n+\frac{2}{3})}-\frac{5}{72}\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{1}{(n+\frac{2}{3})(n+1)}$$$$+\frac{1}{36}\underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{1}{(n+1)(n+\frac{1}{3})}$$$$\mathrm{S}=\frac{1}{24}.\frac{\psi \left(\frac{2}{3}\right)-\psi \left(\frac{1}{3}\right)}{\frac{2}{3}-\frac{1}{3}}-\frac{5}{72}.\frac{\psi \left(1\right)-\psi \left(\frac{2}{3}\right)}{1-\frac{2}{3}}$$$$+\frac{1}{36}.\frac{\psi \left(1\right)-\psi \left(\frac{1}{3}\right)}{1-\frac{1}{3}}$$$$\mathrm{S}=\frac{1}{8}(\psi \left(\frac{2}{3}\right)-\psi \left(\frac{1}{3}\right))-\frac{5}{24}(\psi \left(1\right)-\psi \left(\frac{2}{3}\right))$$$$+\frac{1}{24}(\psi \left(1\right)-\psi \left(\frac{1}{3}\right))$$$$\mathrm{S}=-\frac{1}{6}\psi \left(\frac{1}{3}\right)+\frac{1}{3}\psi \left(\frac{2}{3}\right)-\frac{1}{6}\psi \left(1\right)$$$$$$$$\bullet \psi (1-z)=\psi \left(z\right)+\pi \cot \left(\pi z\right)$$$$\Rightarrow \psi \left(\frac{2}{3}\right)=\psi \left(\frac{1}{3}\right)+\pi \cot \left(\frac{\pi }{3}\right)$$$$\psi \left(\frac{2}{3}\right)=\psi \left(\frac{1}{3}\right)+\frac{\pi }{\sqrt{3}}$$$$\mathrm{S}=\frac{1}{6}\psi \left(\frac{1}{3}\right)+\frac{\pi }{3\sqrt{3}}-\frac{1}{6}\psi \left(1\right)$$$$\bullet \psi \left(1\right)=-\gamma$$$$\bullet \psi \left(\frac{1}{3}\right)=-\frac{\pi }{2\sqrt{3}}-\frac{3}{2}\ln 3-\gamma$$$$\mathrm{S}=\frac{1}{6}(-\frac{\pi }{2\sqrt{3}}-\frac{3}{2}\ln 3-\gamma )+\frac{\pi }{3\sqrt{3}}+\frac{\gamma }{6}$$$$\mathrm{S}=\frac{\pi }{4\sqrt{3}}-\frac{1}{4}\ln 3$$

Commented by Tawa11 last updated on 24/Jul/21

$$\mathrm{Wow},\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.$$

Commented by Tawa11 last updated on 24/Jul/21

$$\mathrm{Sir},\mathrm{where}\mathrm{can}\mathrm{I}\mathrm{learn}\mathrm{summation}\mathrm{like}\mathrm{this}.\mathrm{Please}\mathrm{sir}\mathrm{prescribe}\mathrm{book}.\mathrm{I}\mathrm{want}$$$$\mathrm{to}\mathrm{study}\mathrm{on}\mathrm{it}\mathrm{too}.\mathrm{Thanks}\mathrm{sir}.$$

Commented by Tawa11 last updated on 24/Jul/21

$$\mathrm{Sir}.$$$$\mathrm{Can}\mathrm{we}\mathrm{find}\mathrm{sum}\mathrm{to}\mathrm{n}\mathrm{terms}\mathrm{here}:\frac{1}{1.2.3}+\frac{1}{4.5.6}+\frac{1}{7.8.9}+...$$

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