calculate-u-n-k-1-n-k-k-1-

Question Number 35035 by abdo mathsup 649 cc last updated on 14/May/18

c a l c u l a t e u_{n} = \sum_{k = 1}^{n} \frac{k}{(k + 1)!}

Answered by tanmay.chaudhury50@gmail.com last updated on 14/May/18

T_{k} = \frac{k}{(k + 1)!}

= \frac{k + 1 - 1}{(k + 1)!}

T_{k} = \frac{1}{k!} - \frac{1}{(k + 1)!}

T_{1} = \frac{1}{1!} - \frac{1}{2!}

T_{2} = \frac{1}{2!} - \frac{1}{3!}

T_{3} = \frac{1}{3!} - \frac{1}{4!}

\dots \dots \dots \dots \dots .

\dots \dots \dots \dots \dots

T_{n} = \frac{1}{n!} - \frac{1}{(n + 1)!}

s o T_{1} + T_{2} + T_{3} + \dots . + T_{n} = \frac{1}{1!} - \frac{1}{(n + 1)!}

= 1 - \frac{1}{(n + 1)!}