k = 0 n C k n k + 1 =

Question

$\sum_{k = 0}^{n} \frac{{}^{n}C_{k}}{k + 1} =$

Smart Achievers · Accepted Answer

(3)

$\sum_{k = 0}^{n} \frac{{}^{n}C_{k}}{k + 1} = \sum_{k = 0}^{n} \frac{n!}{(k + 1) k! (n - k)!}$

$= \sum_{k = 0}^{n} \frac{n! (n + 1)}{(k + 1)! (n - k)! (n + 1)}$

$= \frac{1}{n + 1} \sum_{k = 0}^{n} \frac{(n + 1)!}{(k + 1)! (n + 1 - k - 1)!}$

$= \frac{1}{n + 1} \sum_{k = 0}^{n} {}^{n + 1}C_{k + 1} = \frac{1}{n + 1} [2^{n + 1} - 1]$

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