Question Number 2497 by 123456 last updated on 21/Nov/15
$$\mathrm{S}=\underset{{n}=\mathrm{0}} {\overset{+\infty} {\sum}}\frac{{n}+\mathrm{2}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{3}\right)} \\ $$$$\mathrm{does}\:\mathrm{S}\:\mathrm{converge}? \\ $$
Answered by prakash jain last updated on 21/Nov/15
![S=Σ_(n=0) ^∞ ((n+2)/((n+1)(n+3)))=(1/2)Σ_(n=0) ^∞ [(1/(n+1))+(1/(n+3))] S diverges as (1/n) does not converge.](Q2517.png)
$$\mathrm{S}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}+\mathrm{2}}{\left({n}+\mathrm{1}\right)\left({n}+\mathrm{3}\right)}=\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left[\frac{\mathrm{1}}{{n}+\mathrm{1}}+\frac{\mathrm{1}}{{n}+\mathrm{3}}\right] \\ $$$$\mathrm{S}\:\mathrm{diverges}\:\mathrm{as}\:\frac{\mathrm{1}}{{n}}\:\mathrm{does}\:\mathrm{not}\:\mathrm{converge}. \\ $$