How-do-I-find-the-sum-of-1-3x-6x-2-10x-3-15x-4-where-1-lt-x-lt-1-Please-Help-

Question Number 136161 by nimnim last updated on 19/Mar/21

H o w d o I f i n d t h e s u m o f

1 + 3 x + 6 x^{2} + 10 x^{3} + 15 x^{4} + \dots \dots . \infty, w h e r e - 1 < x < 1 ?

P l e a s e H e l p . .

Answered by bramlexs22 last updated on 19/Mar/21

S = 1 + 3 x + 6 x^{2} + 10 x^{3} + 15 x^{4} + \dots

x S = x + 3 x^{2} + 6 x^{3} + 10 x^{4} + \dots

(1 - x) S = 1 + 2 x + 3 x^{2} + 4 x^{3} + 5 x^{4} + \dots

l e t T = 1 + 2 x + 3 x^{2} + 4 x^{3} + 5 x^{4} + \dots

\int T d x = x + x^{2} + x^{3} + x^{4} + x^{5} + \dots

\int T d x = \frac{x}{1 - x} t h e n T = \frac{1}{{(1 - x)}^{2}}

s o S = \frac{1}{{(1 - x)}^{3}}

Commented by nimnim last updated on 19/Mar/21

T h a n k y o u S i r .

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