1. Tardigrade
  2. Question
  3. Mathematics
  4. The sum to infinity of the series 1+2(1-(1/n)) +3( 1-(1/n))2 + ..... where n ∈ N, is given by

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The sum to infinity of the series $1+2\left(1-\frac{1}{n}\right) +3\left( 1-\frac{1}{n}\right)^{2} + ..... $ where $n \in N$, is given by

Sequences and Series

Solution:

Let $S = 1+2 \left(1-\frac{1}{n}\right) + 3\left(1-\frac{1}{n}\right)^{2} + .....\quad...\left(i\right)$
$ \therefore \left(1-\frac{1}{n}\right)S$
$= \left(1-\frac{1}{n}\right) +2\left(1-\frac{1}{n}\right)^{2} + ..... \quad...\left(ii\right)$
$ \left(i\right)-\left(ii\right)$ gives
$\frac{S}{n} = 1+\left(1-\frac{1}{n}\right)+\left(1-\frac{1}{n}\right)^{2} + .....\infty $
$ = \frac{1}{1-\left(1-\frac{1}{n}\right)} = n $
$ \Rightarrow S=n^{2}$