First slide

Binomial theorem for negative integral and rational Index

Question

$\sum_{k = 1}^{\infty} k {(1 - \frac{1}{n})}^{k - 1} =$

Moderate

A

n(n - 1)
B

n(n+ l)
C

n²
D

(n + 1)²

Solution

$\sum_{k = 1}^{\infty} k {(1 - \frac{1}{n})}^{k - 1}$
$\begin{matrix} = 1 + 2 {(1 - \frac{1}{n})}^{1} + 3 {(1 + \frac{1}{n})}^{2} + \dots \\ = 1 + 2 t + 3 t^{2} + \dots \\ = (1 - t)^{- 2} \\ = {[1 - (1 - \frac{1}{n})]}^{- 2} = {(\frac{1}{n})}^{- 2} = n^{2} \end{matrix}$

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