Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 116480 by Dwaipayan Shikari last updated on 04/Oct/20

(1/8)+(1/(18))+(1/(30))+(1/(44))+(1/(60))+(1/(78))+(1/(98))+(1/(120))+........

$$\frac{\mathrm{1}}{\mathrm{8}}+\frac{\mathrm{1}}{\mathrm{18}}+\frac{\mathrm{1}}{\mathrm{30}}+\frac{\mathrm{1}}{\mathrm{44}}+\frac{\mathrm{1}}{\mathrm{60}}+\frac{\mathrm{1}}{\mathrm{78}}+\frac{\mathrm{1}}{\mathrm{98}}+\frac{\mathrm{1}}{\mathrm{120}}+........ \\ $$

Answered by Olaf last updated on 04/Oct/20

u_n  = (1/(n^2 +7n)) = (1/(n(n+7))) = (1/7)((1/n)−(1/(n+7)))  Σ_(n=1) ^∞ u_n  = (1/7)(Σ_(n=1) ^∞ (1/n)−Σ_(n=1) ^∞ (1/(n+7)))  Σ_(n=1) ^∞ u_n  = (1/7)(Σ_(n=1) ^∞ (1/n)−Σ_(n=8) ^∞ (1/n))  Σ_(n=1) ^∞ u_n  = (1/7)Σ_(n=1) ^7 (1/n) = (1/7)×((363)/(140)) = ((363)/(980))

$${u}_{{n}} \:=\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} +\mathrm{7}{n}}\:=\:\frac{\mathrm{1}}{{n}\left({n}+\mathrm{7}\right)}\:=\:\frac{\mathrm{1}}{\mathrm{7}}\left(\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{{n}+\mathrm{7}}\right) \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{u}_{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{7}}\left(\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}}−\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}+\mathrm{7}}\right) \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{u}_{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{7}}\left(\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}}−\underset{{n}=\mathrm{8}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}}\right) \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}{u}_{{n}} \:=\:\frac{\mathrm{1}}{\mathrm{7}}\underset{{n}=\mathrm{1}} {\overset{\mathrm{7}} {\sum}}\frac{\mathrm{1}}{{n}}\:=\:\frac{\mathrm{1}}{\mathrm{7}}×\frac{\mathrm{363}}{\mathrm{140}}\:=\:\frac{\mathrm{363}}{\mathrm{980}} \\ $$$$ \\ $$

Commented by Dwaipayan Shikari last updated on 04/Oct/20

Great sir!

$$\mathrm{Great}\:\mathrm{sir}! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com