Provide an appropriate response.
-Find the sum of the series $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n - 1 } }$ by expressing $\frac { 1 } { 1 - x }$ as a geometric series, differentiating both sides of the resulting equation with respect to $x$, multiplying both sides by $x$, differentiating again, and replacing $x$ by $\frac { 1 } { 3 }$.
A) 6
B) 9
C) $\frac { 3 } { 2 }$
D) $\frac { 9 } { 2 }$

Question

Quizplus · Accepted Answer

We start by expressing $\frac { 1 } { 1 - x }$ as a geometric series:
$$\frac { 1 } { 1 - x } = \sum_{n=0}^{\infty}x^n$$
Differentiating both sides with respect to $x$ gives us:
$$\frac { 1 } { (1 - x)^2 } = \sum_{n=1}^{\infty}nx^{n-1}$$
Multiplying both sides by $x$ gives us:
$$\frac { x } { (1 - x)^2 } = \sum_{n=1}^{\infty}nx^{n}$$
Differentiating again with respect to $x$ gives us:
$$\frac { 1 + 3x } { (1 - x)^3 } = \sum_{n=1}^{\infty}n^2 x^{n-1}$$
Now we replace $x$ with $\frac {1} {3}$:
$$\frac { 9 } { 8 } = \sum_{n=1}^{\infty}\frac { n ^ { 2 } } { 3 ^ { n - 1 } }$$
Therefore, the sum of the given series is $\frac { 9 } { 2 }$, which corresponds to option D.

Provide an Appropriate Response ∑n=1∞n23n−1\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n - 1 } }∑n=1∞​3n−1n2​

Provide an Appropriate Response $\sum _ { n = 1 } ^ { \infty } \frac { n ^ { 2 } } { 3 ^ { n - 1 } }$