For each positive integer n, let sn=31.2.4+42.3.5+53.4.6+…..+n+2n(n+1)(n+3)Then limn→∞ sn equals

Let uk=k+2k(k+1)(k+3)=(k+2)2k(k+1)(k+2)(k+3) =k2+4k+4k(k+1)(k+2)(k+3) =k(k+1)+3k+4k(k+1)(k+2)(k+3)=1(k+2)(k+3)+3(k+1)(k+2)(k+3)+4k(k+1)(k+2)(k+3) =1k+2-1k+3-321(k+2)(k+3)-1(k+1)(k+2)-431(k+1)(k+2)(k+3)-1k(k+1)(k+2)Now, put k=1,2,3,…,n and add. Thus su=u1+u2+…..+un =13-1n+3-321(n+2)(n+3)-12.3-431(n+1)(n+2)(n+3)-11.2.3Therefore limn→∞ sn=13+312+418=2936

For each positive integer n, let sn=31.2.4+42.3.5+53.4.6+…..+n+2n(n+1)(n+3)Then limn→∞ sn equals

Let uk=k+2k(k+1)(k+3)=(k+2)2k(k+1)(k+2)(k+3) =k2+4k+4k(k+1)(k+2)(k+3) =k(k+1)+3k+4k(k+1)(k+2)(k+3)=1(k+2)(k+3)+3(k+1)(k+2)(k+3)+4k(k+1)(k+2)(k+3) =1k+2-1k+3-321(k+2)(k+3)-1(k+1)(k+2)-431(k+1)(k+2)(k+3)-1k(k+1)(k+2)Now, put k=1,2,3,…,n and add. Thus su=u1+u2+…..+un =13-1n+3-321(n+2)(n+3)-12.3-431(n+1)(n+2)(n+3)-11.2.3Therefore limn→∞ sn=13+312+418=2936

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For each positive integer n, let $s_{n} = \frac{3}{1 .2 . 4} + \frac{4}{2 .3 . 5} + \frac{5}{3 .4 . 6} + \dots . . + \frac{n + 2}{n (n + 1) (n + 3)}$ Then $\lim_{n \to \infty} s_{n}$ equals

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$\frac{29}{6}$

$\frac{29}{18}$

$\frac{29}{36}$

answer is B.

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Detailed Solution

Let $u_{k} = \frac{k + 2}{k (k + 1) (k + 3)} = \frac{(k + 2)^{2}}{k (k + 1) (k + 2) (k + 3)} = \frac{k^{2} + 4 k + 4}{k (k + 1) (k + 2) (k + 3)}$ $\begin{array}{r} = \frac{k (k + 1) + 3 k + 4}{k (k + 1) (k + 2) (k + 3)} = \frac{1}{(k + 2) (k + 3)} + \frac{3}{(k + 1) (k + 2) (k + 3)} + \frac{4}{k (k + 1) (k + 2) (k + 3)} \\ = (\frac{1}{k + 2} - \frac{1}{k + 3}) - \frac{3}{2} [\frac{1}{(k + 2) (k + 3)} - \frac{1}{(k + 1) (k + 2)}] - \frac{4}{3} [\frac{1}{(k + 1) (k + 2) (k + 3)} - \frac{1}{k (k + 1) (k + 2)}] \end{array}$ Now, put $k = 1,2, 3, \dots, n$ and add. Thus $s_{u} = u_{1} + u_{2} + \dots . . + u_{n}$

$= (\frac{1}{3} - \frac{1}{n + 3}) - \frac{3}{2} [\frac{1}{(n + 2) (n + 3)} - \frac{1}{2 .3}] - \frac{4}{3} [\frac{1}{(n + 1) (n + 2) (n + 3)} - \frac{1}{1 .2 . 3}]$ Therefore $\lim_{n \to \infty} s_{n} = \frac{1}{3} + \frac{3}{12} + \frac{4}{18} = \frac{29}{36}$