# Sum of n terms of series $\;\frac{1}{1^3}\;+\frac{1+2}{1^3+2^3}\;+\frac{1+2+3}{1^3+2^3+3^3}\;+....\;is$

$(a)\;\frac{n}{n+1}\qquad(b)\;\frac{2n}{n+1}\qquad(c)\;\frac{n}{2(n+1)}\qquad(d)\;\frac{2}{n(n+1)}$

Answer : (b) $\;\frac{2n}{n+1}$
Explanation : $t_{n}=\frac{1+2+3+\;....\;+n}{1^3+2^3+3^3+\;....\;+n^3}=\frac{\frac{n(n+1)}{2}}{(\frac{n(n+1)}{2})^2}$
$t_{n}=\frac{2}{n(n+1)}=2\;[\frac{1}{n}-\frac{1}{n+1}]$
$\sum\;t_{n}=2\;[(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+\;...\;(\frac{1}{n}-\frac{1}{n+1})]$
$S_{n}=2\;[1-\frac{1}{n+1}]=\frac{2n}{n+1}.$
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