Question

i need to show whether the series is convergent or divergent by expressing Sn as a telescoping sum. If it is convergent, find its sum. \[\sum_{1}^{\infty}2\div \left( n ^{2}+4n+3 \right)\]

Answer 1

\[\sum_{1}^{\infty}\frac 2{(n+1)(n+3)}=\sum_{1}^{\infty}\frac {(n+3)-(n+1)}{(n+1)(n+3)}\]\[t_n=\frac1{n+1}-\frac1{n+3}\]\[t_1=1/2-1/4\]\[t_2=1/3-1/5\]\[t_3=1/4-1/6\]\[\S_n=1/2+1/3-1/(n+2)-1/(n+3)\] when n->infinity\[s_n=1/2+1/3=5/6\] so convergent