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a-Find-the-sum-given-by-S-n-1-1-3-1-3-5-1-5-7-1-2n-1-2n-1-b-find-the-limit-of-S-n-as-n-




Question Number 8764 by tawakalitu last updated on 26/Oct/16
(a) Find the sum given by  S_n  = (1/(1.3)) + (1/(3.5)) + (1/(5.7)) + ... + (1/((2n − 1)(2n + 1)))  (b) find the limit of   S_n   as  n → ∞
$$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{given}\:\mathrm{by} \\ $$$$\mathrm{S}_{\mathrm{n}} \:=\:\frac{\mathrm{1}}{\mathrm{1}.\mathrm{3}}\:+\:\frac{\mathrm{1}}{\mathrm{3}.\mathrm{5}}\:+\:\frac{\mathrm{1}}{\mathrm{5}.\mathrm{7}}\:+\:…\:+\:\frac{\mathrm{1}}{\left(\mathrm{2n}\:−\:\mathrm{1}\right)\left(\mathrm{2n}\:+\:\mathrm{1}\right)} \\ $$$$\left(\mathrm{b}\right)\:\mathrm{find}\:\mathrm{the}\:\mathrm{limit}\:\mathrm{of}\:\:\:\mathrm{S}_{\mathrm{n}} \:\:\mathrm{as}\:\:\mathrm{n}\:\rightarrow\:\infty \\ $$
Commented by sou1618 last updated on 26/Oct/16
S_n =(1/2){((1/1)−(1/3))+((1/3)−(1/5))+....+((1/(2n−1))−(1/(2n+1)))}    =(1/2)(1−(1/(2n+1)))    =(n/(2n+1))  lim_(n→∞) (n/(2n+1))=lim_(n→∞) (1/( (2+(1/n)) ))=(1/2)
$${S}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\left\{\left(\frac{\mathrm{1}}{\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{3}}\right)+\left(\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{5}}\right)+….+\left(\frac{\mathrm{1}}{\mathrm{2}{n}−\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\right)\right\} \\ $$$$\:\:=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{1}}\right) \\ $$$$\:\:=\frac{{n}}{\mathrm{2}{n}+\mathrm{1}} \\ $$$${lim}_{{n}\rightarrow\infty} \frac{{n}}{\mathrm{2}{n}+\mathrm{1}}={lim}_{{n}\rightarrow\infty} \frac{\mathrm{1}}{\:\left(\mathrm{2}+\frac{\mathrm{1}}{{n}}\right)\:}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$
Commented by tawakalitu last updated on 26/Oct/16
Thank you sir. God bless you.
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{God}\:\mathrm{bless}\:\mathrm{you}. \\ $$

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