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

Question

The sum to infinity of the series 1+2(1-(1/n)) +3( 1-(1/n))2 + ..... where n ∈ N, is given by (A) n(n-1) (B) n(1-(1/n))2 (C)  n2 (D) ((n-1/n))2

Tardigrade Admin · Accepted Answer

Let S = 1+2 (1-(1/n)) + 3(1-(1/n))2 + ..... ...(i) ∴ (1-(1/n))S = (1-(1/n)) +2(1-(1/n))2 + ..... ...(ii) (i)-(ii) gives (S/n) = 1+(1-(1/n))+(1-(1/n))2 + .....∞ = (1/1-(1-(1)n)) = n ⇒ S=n2

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

$n (n - 1)$

$n (1 - \frac{1}{n})^{2}$

$n^{2}$

$(\frac{n - 1}{n})^{2}$

Q. The sum to infinity of the series 1+2(1−n1​)+3(1−n1​)2+..... where n∈N, is given by

n(n−1)

n(1−n1​)2

n2

(nn−1​)2

Let S=1+2(1−n1​)+3(1−n1​)2+........(i) ∴(1−n1​)S =(1−n1​)+2(1−n1​)2+........(ii) (i)−(ii) gives nS​=1+(1−n1​)+(1−n1​)2+.....∞ =1−(1−n1​)1​=n ⇒S=n2

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

$n (n - 1)$

$n (1 - \frac{1}{n})^{2}$

$n^{2}$

$(\frac{n - 1}{n})^{2}$