If |x|

Question

(D) Let $S = 1 + 2x + 3x^2 + 4x^3 + \dots \infty$. Multiplying by $x$,we get $xS = x + 2x^2 + 3x^3 + \dots \infty$. Subtracting the two equations: $S

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

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(D) Let $S = 1 + 2x + 3x^2 + 4x^3 + \dots \infty$. Multiplying by $x$,we get $xS = x + 2x^2 + 3x^3 + \dots \infty$. Subtracting the two equations: $S - xS = 1 + (2x - x) + (3x^2 - 2x^2) + (4x^3 - 3x^3) + \dots \infty$. $S(1 - x) = 1 + x + x^2 + x^3 + \dots \infty$. The right side is an infinite geometric series with first term $a = 1$ and common ratio $r = x$. Since $|x| Therefore,$S(1 - x) = \frac{1}{1 - x}$. $S = \frac{1}{(1 - x)^2}$.

If $|x| < 1$,what is the sum of the series $1 + 2x + 3x^2 + 4x^3 + \dots \infty$?

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