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Question Number 157576 by amin96 last updated on 24/Oct/21

Commented by quvonch3737 last updated on 24/Oct/21

$1 + \frac{1}{x} + \frac{9}{x^{2}} + \frac{16}{x^{3}} + . . . = S / \times \frac{1}{x}$ $\frac{1}{x} + \frac{4}{x^{2}} + \frac{9}{x^{3}} + \frac{16}{x^{4}} + . . . = \frac{S}{x}$ $S - \frac{S}{x} = 1 + \frac{3}{x} + \frac{5}{x^{2}} + \frac{7}{x^{3}} + \frac{9}{x^{4}} + . . . = 91$ $S (1 - \frac{1}{x}) = 91$ $S = \frac{91 x}{x - 1 r}$
Commented by Rasheed.Sindhi last updated on 25/Oct/21

$S l i g h t l y d e f f e r e n t w a y . . .$ $1 + \frac{3}{x} + \frac{5}{x^{2}} + \frac{7}{x^{3}} + \frac{9}{x^{4}} + . . . = 91$ $1 + \frac{4}{x} + \frac{9}{x^{2}} + \frac{16}{x^{3}} + \frac{25}{x^{4}} + . . . = S$ $S - 91 = \frac{1}{x} + \frac{4}{x^{2}} + \frac{9}{x^{3}} + \frac{16}{x^{4}} + . . .$ $= \frac{1}{x} (1 + \frac{4}{x} + \frac{9}{x^{2}} + \frac{16}{x^{3}} + . . .)$ $= \frac{1}{x} (S)$ $S - \frac{S}{x} = 91$ $S (\frac{x - 1}{x}) = 91$ $S = \frac{91 x}{x - 1}$
	Answered by mr W last updated on 25/Oct/21

	$l e t t = \frac{1}{x}$ $S_{1} = 1 + 3 t + 5 t^{2} + 7 t^{3} + 9 t^{4} + . . . = 91$ ${t S}_{1} = t + 3 t^{2} + 5 t^{3} + 7 t^{4} + 9 t^{5} + . . .$ $(1 - t) S_{1} = 1 + 2 (t + t^{2} + t^{3} + t^{4} + . . .)$ $(1 - t) S_{1} = - 1 + \frac{2}{1 - t} (o n l y i f ∣ t ∣< 1, i . e . ∣ x ∣> 1)$ $\frac{2}{{(1 - t)}^{2}} - \frac{1}{1 - t} - S_{1} = 0$ $\Rightarrow \frac{1}{1 - t} = \frac{1 \pm \sqrt{1 + 8 S_{1}}}{4}$ $= \frac{1 \pm \sqrt{1 + 8 \times 91}}{4} = \frac{1 \pm 27}{4} = 7, - \frac{13}{2}$ $∣ t ∣=∣ \frac{1}{x} ∣< 1$ $- 1 < t < 1$ $\Rightarrow 0 < 1 - t < 2$ $\Rightarrow \frac{1}{1 - t} > \frac{1}{2}$ $\Rightarrow o n l y \frac{1}{1 - t} = 7 i s v a l i d, - \frac{13}{2} i s r e j e c t e d$ $i . e . x = \frac{1}{t} = \frac{7}{6}$ $S_{2} = 1 + \frac{4}{x} + \frac{9}{x^{2}} + \frac{16}{x^{3}} + \frac{25}{x^{4}} + . . .$ $S_{2} = 1 + 4 t + 9 t^{2} + 16 t^{3} + 25 t^{4} +$ $S_{1} = 1 + 3 t + 5 t^{2} + 7 t^{3} + 9 t^{4} + . . .$ $S_{2} - S_{1} = t + 4 t^{2} + 9 t^{3} + 16 t^{4}$ $\frac{S_{2} - S_{1}}{t} = 1 + 4 t + 9 t^{2} + 16 t^{3} + . . .$ $\frac{S_{2} - S_{1}}{t} = S_{2}$ $S_{2} = \frac{S_{1}}{1 - t} = 91 \times 7 = 637$ $\Rightarrow 1 + \frac{4}{x} + \frac{9}{x^{2}} + \frac{16}{x^{3}} + \frac{25}{x^{4}} + . . . = 637$
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