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Question Number 200931 by Spillover last updated on 26/Nov/23

$$\frac{1}{9}+\frac{1}{18}+\frac{1}{30}+\frac{1}{45}+\frac{1}{63}+\frac{1}{84}+...\mathrm{\infty }=?$$$$$$

Answered by MM42 last updated on 26/Nov/23

$$s_n=\frac{2}{3}\left(\frac{1}{(n+1)(n+2)}\right)=\frac{2}{3}(\frac{1}{n+1}-\frac{1}{n+2})$$$$\Rightarrow s_n=\frac{2}{3}(\frac{1}{2}-\frac{1}{n+2})\stackrel{n\to \mathrm{\infty }}{\Rightarrow }s=\frac{1}{3}✓$$$$$$$$$$

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