1+2x+3x2+4x3 to infinity sum of this series - Maths - Sequences and Series

Question

1+2x+3x2+4x3 to infinity
sum of this series

Vipin Verma · Accepted Answer

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 )

1+2x+3x2+4x3..........to infinity sum of this series

1+2x+3x²+4x³..........to infinity

sum of this series