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1-9-1-18-1-30-1-45-1-63-1-84-




Question Number 200931 by Spillover last updated on 26/Nov/23
(1/9)+(1/(18))+(1/(30))+(1/(45))+(1/(63))+(1/(84))+...∞=?
$$\frac{\mathrm{1}}{\mathrm{9}}+\frac{\mathrm{1}}{\mathrm{18}}+\frac{\mathrm{1}}{\mathrm{30}}+\frac{\mathrm{1}}{\mathrm{45}}+\frac{\mathrm{1}}{\mathrm{63}}+\frac{\mathrm{1}}{\mathrm{84}}+…\infty=? \\ $$$$ \\ $$
Answered by MM42 last updated on 26/Nov/23
s_n =(2/3)((1/((n+1)(n+2))))=(2/3)((1/(n+1))−(1/(n+2)))    ⇒s_n =(2/3)((1/2)−(1/(n+2))) ⇒^(n→∞)  s=(1/3)  ✓
$${s}_{{n}} =\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)}\right)=\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{1}}{{n}+\mathrm{1}}−\frac{\mathrm{1}}{{n}+\mathrm{2}}\right)\:\: \\ $$$$\Rightarrow{s}_{{n}} =\frac{\mathrm{2}}{\mathrm{3}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{{n}+\mathrm{2}}\right)\:\overset{{n}\rightarrow\infty} {\Rightarrow}\:{s}=\frac{\mathrm{1}}{\mathrm{3}}\:\:\checkmark \\ $$$$ \\ $$$$ \\ $$

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