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WZ
Numerade Educator

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Problem 40 Hard Difficulty

(a) Starting with the geometric series $ \sum_{n = 0}^{\infty} x^n, $ find the sum of the series
$ \sum_{n = 1}^{\infty} nx^{n - 1} \mid x \mid < 1 $
(b) Find the sum of each of the following series.
(i) $ \sum_{n = 1}^{\infty} nx^n, \mid x \mid < 1 $
(ii) $ \sum_{n = 1}^{\infty} \frac {n}{2^n} $
(c) Find the sum of each of the following series.
(i) $ \sum_{n = 2}^{\infty} n(n - 1) x^n, \mid x \mid < 1 $
(ii) $ \sum_{n = 2}^{\infty} \frac {n^2 - n}{2^n} $
(iii) $ \sum_{n = 1}^{\infty} \frac {n^2}{2^n} $

Answer

A. $\frac{1}{(1-x)^{2}}, \quad|x|<1$
B. (i) $\frac{x}{(1-x)^{2}}, \quad|x|<1, \quad$ (ii) 2
C. (i) $\frac{2 x^{2}}{(1-x)^{3}}$
(ii) 4
(iii) 6

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Video Transcript

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