1+2x+3x2+4x3..........to infinity

sum of this series

Let : S = 1 + 2x + 3x² + 4x³ + 5x4 + ......... (1)

....Then : x.S = x + 2x² + 3x³ + 4x4 + ..........(2) ( multiply both sides by x)
.........................................

Then, ... Eq(1) - Eq(2) ... gives

S - xS = 1 + x + x² + x³ + ...

S ( 1 - x ) = 1 + x + x² + x³ + ...
The expression on the right hand sides is infinite GP of a = 1, r = x
So the sum of this GP is 1/(1-x)
Or S ( 1 - x ) = 1/(1-x)
S = 1 / ( 1 - x )² ............(Ans )

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