Show that 1 - 2 x - 3 x 2 - 4 x 3 - - - - 3 - 2 - 1 - 3 x - 6 x 2 - 10 x 3 - - - - - where - x - - 1

[ ( 1+2x+3 x^ 2 +4 x^ 3 +.......∞ #160; ) ^ 3/2 =1+3x+6 x^ 2 +......∞ #160;; ...

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Summation by Sigma Method

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Question

Show that ${(1 + 2 x + 3 x^{2} + 4 x^{3} + \dots \dots . \infty)}^{3 / 2} = 1 + 3 x + 6 x^{2} + 10 x^{3} + \dots \dots, \infty,$ where $| x | < 1$

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Solution

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Similar questions

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

x \neq 1

f (x) = {cot}^{- 1} (\frac{1 - x^{2}}{2 x}) + {cot}^{- 1} (\frac{1 - 3 x^{2}}{3 x - x^{3}}) - {cot}^{- 1} (\frac{1 - 6 x^{2} + x^{4}}{4 x - 4 x^{3}})

F^{'} (x)

1 + 2 x + 3 x^{2} + 4 x^{3} + . . . ., | x | < 1

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2 \times 1 + 3 \times 2 + 4 \times 4 + 5 \times 8 + . . . .

x^{24}

{(1 + 3 x + 6 x^{2} + 10 x^{3} + . . . . . . . . \infty)}^{2 / 3}

1 + 2 x + 3 x^{2} + 4 x^{3} + . . .

| x | < 1.

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