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If $|x| \lt 1$, then the sum of the series $1+2 x+3 x^2+4 x^3+\ldots \ldots \ldots . . \infty$ will be
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Verified Answer
The correct answer is:
$\frac{1}{(1-x)^2}$
This is an A.G.P.
Let $S=1+2 x+3 x^2+\ldots \ldots \infty$
$\Rightarrow x . S=x+2 x^2+\ldots \ldots \infty$
Subtracting $(1-x) S=1+x+$ $x^2+\ldots \ldots \ldots \infty=\frac{1}{1-x}$
$\therefore S=\frac{1}{(1-x)^2}$
Aliter: use $S=\left[1+\frac{r}{1-r} \times\right.$ diff. of A.P. $] \frac{1}{1-r}$
Let $S=1+2 x+3 x^2+\ldots \ldots \infty$
$\Rightarrow x . S=x+2 x^2+\ldots \ldots \infty$
Subtracting $(1-x) S=1+x+$ $x^2+\ldots \ldots \ldots \infty=\frac{1}{1-x}$
$\therefore S=\frac{1}{(1-x)^2}$
Aliter: use $S=\left[1+\frac{r}{1-r} \times\right.$ diff. of A.P. $] \frac{1}{1-r}$
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