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# (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) $\sum_{n = 1}^{\infty} \frac {n^2}{2^n}$

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

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The problem is starting ways that Jemaah trick Siri's summon from zero to infinity backs to the power of in blinded arm of the Siri's, some from wandering Trinity in Tom's extra months, one at volume fax is the last one. So first, some off from zero to Infinity Axe to end. This is Echo one over one, minus tax. I have absolutely no fax is That's the one. So half the sum from one to infinity E X was one cams, and this is a cartoon determinative of this Siri's. This is equal to the derivative of the function. One over one man is sex. This is a call to one over one minus X square. You're absolutely our backs. It's unless what heart be finding the thumb of each of the following Siri's one, some from one to infinity and hams acts to end. This is just axe toms, some from one to twenty in town sax, too, and minus one. So we use the result in part A. So we have. This is equal to fax over one minus x square. You're absolutely Fax lesson one. Party two services, some from wanting twenty and over to two of them. So we just replace X by half. We can't some of from one to twenty on over to end. This is a caught you well, half over. One man is my half stoy mentions the quanto too hard to see finding the thumb of each of the following Siri's one. So for the first one we can write this's X X squared hams, some round two to entity and House months. One ham's axe, too and minus two. This is a hot X square harms the derivative of the theories. I'm from wanting unity in times Axe, too and minus one. Just let me talk to Ex Squire Toms. Zero of the function one over one minus X squared is out There is to explain over one month's axe to the power of three. Heart is too, for part of two would just replace exp I want Huff. We have some from to Infinity and Squire months and over to end. Serious is equal to to Pam's behalf Squire over one minus my half. It's a power of three. The test is the record, too. Four part of it. Three. What this Siri's This is just but the reason Siri's is equal to the Siri's from to infinity and squire months and over two to attend class Siri's and over two to ten from party, too. We have this party is equal to two from part of C. To have this party is equal to four. The answer is six.

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