IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

sunnyguha

$\sum_{r=1}^{n}r{x}^{r-1}$ plz solve it only by calculus

shikhar_destructo

we know 1+x+x^2+x^3+.......X^n= (x^n-1)/(x-1) DIFF BOTH THE SIDES AND GET THE REQD RESULT. HOPE IT HELPS.

magiclko

$\Sigma r {x}^{r-1}$ = x + 2x + 3x² + 4x³ + ..... nx^n-1

u knw that 1+x +x² + x³ +... + xⁿ = 1-xⁿ⁺¹/ 1-x

differentitaing, u'll get the answer,

x + 2x + 3x² + 4x³ + ..... nx^n-1 = {(xⁿ⁺¹-1) / (1-x²) } - {(n+1)xⁿ / (1-x)}