$y = 3 x + 6 x^{2} + 10 x^{3} + . . . . . . . \infty$ হলে, দেখাও যে, $x = \frac{1}{3} y - \frac{1}{y^{2}} . \frac{4}{2!} y^{2} + \frac{1.4.7}{3^{3} . 3!} y^{3} - . . . . . . . . . .$

Created: 1 year ago | Updated: 1 year ago

Updated: 1 year ago

$y = 3 x + 6 x^{2} + 10 x^{3} + . . . . . . . . . . . . . . .$

$\Rightarrow 1 + y = 1 + 3 x + 6 x^{2} + 10 x^{3} + . . . = {(1 - x)}^{- 3}$

$\Rightarrow {(1 + y)}^{- \frac{1}{3}} = {(1 - x)}^{- 3 \times (- \frac{1}{3})}$

$\Rightarrow 1 - x = {(1 + y)}^{- \frac{1}{3}}$

$\Rightarrow 1 - x = 1 - \frac{1}{3} y + \frac{(- \frac{1}{3}) (- \frac{1}{3} - 1)}{2!} y^{2} + \frac{(- \frac{1}{3}) (- \frac{1}{3} - 1) (- \frac{1}{3} - 2)}{3!} y^{3} + . . .$

অতএব, $x = \frac{1}{3} y - \frac{1}{3^{2}} . \frac{4}{2!} y^{2} + \frac{1.4.7}{3^{3} . 3!} y^{3}_. . . . (s h o w e d)$

1 year ago

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$y = 3 x + 6 x^{2} + 10 x^{3} + . . . . . . . \infty$ হলে, দেখাও যে, $x = \frac{1}{3} y - \frac{1}{y^{2}} . \frac{4}{2!} y^{2} + \frac{1.4.7}{3^{3} . 3!} y^{3} - . . . . . . . . . .$

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$y = 3 x + 6 x^{2} + 10 x^{3} + . . . . . . . \infty$ হলে, দেখাও যে, $x = \frac{1}{3} y - \frac{1}{y^{2}} . \frac{4}{2!} y^{2} + \frac{1.4.7}{3^{3} . 3!} y^{3} - . . . . . . . . . .$