calculate-A-n-k-0-n-k-k-1-

Question Number 64444 by mathmax by abdo last updated on 18/Jul/19

c a l c u l a t e A_{n} = \sum_{k = 0}^{n} \frac{k}{(k + 1)!}

Commented by Prithwish sen last updated on 18/Jul/19

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

Commented by mathmax by abdo last updated on 22/Jul/19

A_{n} = \sum_{k = 0}^{n} \frac{k + 1 - 1}{(k + 1)!} = \sum_{k = 0}^{n} (\frac{1}{k!} - \frac{1}{(k + 1)!}) l e t u_{n} = \frac{1}{n!} \Rightarrow

A_{n} = \sum_{k = 0}^{n} (u_{k} - u_{k + 1}) = u_{0} - u_{1} + u_{1} - u_{2} + \dots . + u_{n} - u_{n + 1}

= u_{0} - u_{n + 1} = 1 - \frac{1}{(n + 1)!}

s o A_{n} = 1 - \frac{1}{(n + 1)!} .