IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

sunnyguha

1.-- $\sum_{r=1}^{n} r{x}^{r-1}$

2.-- $\sum_{n=1}^{\infty}{n}^{2}{x}^{n-1}$ |x|<1

PLz guys I need these to be solved only by calcus, will rate all those who answer.

THnx am waiting

ramkumar_november

for the first problem the required sum is.....

$S = 1+2x+3x^2+4x^3+.....+nx^{n-1}$

now consider

T = x+x^2+x^3+x^4+....+x^n

using sum of GP formula

T = x+x^2+x^3+x^4+....+x^n = $\frac{x(x^n-1)}{x-1}$

now find $\frac{dT}{dx}$ which will give you S

zorro

for second one..

consider the series

S = x + 2x^2 + 3x^3 + 4x^4 ----------- infinity
its an AGP, solve to get

x + 2x^2 + 3x^3 + 4x^4 ----------- infinity = x/(1-x)^2

differentiate both sides

1 + 2^2.x + 3^2.x^2 + 4^.x^3----------infinity = d/dx[x/(1-x)^2]

see, LHS is what u want

now solve RHS to get ur answer

cheers !

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