Question

Compute
\[\frac{3}{1\cdot 2\cdot 3}+\frac{5}{2\cdot 3\cdot 4}+\frac{7}{3\cdot 4\cdot 5}+\cdots\]

Answer 1

I think the limit would converge to 1 or thereabouts.

Answer 2

the general term of the expression is (2n+1)/(n(n+1)(n+2)
= (n+(n+1))/(n(n+1)(n+2)
then splitting the terms ,
=1/((n+1)(n+2)) + 1/(n(n+2))
then you can find the two individual summations to n terms then apply limit n tends to infinity.