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Question Number 95145 by i jagooll last updated on 23/May/20

$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{sum}\mathrm{of}$$$$\frac{1}{14}+\frac{1}{35}+\frac{1}{65}+\frac{1}{104}+...?$$

Answered by bobhans last updated on 23/May/20

$$\mathrm{S}=\underset{\mathrm{n}=1}{\sum ^{\mathrm{\infty }}}\left(\frac{2}{(3\mathrm{n}+1)(3\mathrm{n}+2)}\right)=\frac{2}{3}\underset{x\to \mathrm{\infty }}{\lim }\underset{\mathrm{n}=1}{\sum ^{\mathrm{x}}}(\frac{1}{3\mathrm{n}+1}-\frac{1}{3\mathrm{n}+4})$$$$=\frac{2}{3}\underset{x\to \mathrm{\infty }}{\lim }[\frac{1}{4}-\frac{1}{3\mathrm{x}+4}]=\frac{2}{3}\times \frac{1}{4}=\frac{1}{6}$$

Commented by i jagooll last updated on 23/May/20

$$\mathrm{thank}\mathrm{you}$$

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