Show that- 1-2x-3x2-4x3-12-11-x for x 1-

We have, #160; (1 x)^ 2=1+2x+3x^2+4x^3+......... [ Standard representation] for x 1 .Now, (1+2x+3x^2+4x^3+........)^12=(1 x)^ 1 or, (1+2x+ ...

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Greatest Binomial Coefficients

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Question

Show that, $(1 + 2 x + 3 x^{2} + 4 x^{3} + . . . . . .)^{\frac{1}{2}} = \frac{1}{(1 - x)}$ for $x \neq 1$ .

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Solution

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{(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,

| x | < 1

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(| x | < 1)

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

\int (1 + 2 x + 3 x^{2} + 4 x^{3} + . . . . .) d x (| x | < 1)

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