If 0<y<21/3 and x(y3-1)=1 then 2x+23x3+25x5+…………∞

x y3-1=1Given eqn=2 1x+1x33+1x55+..... =log 1+1x1-1x =log 1+(y3-1)1-(y3-1) =log y32-y3

If 0<y<21/3 and x(y3-1)=1 then 2x+23x3+25x5+…………∞

x y3-1=1Given eqn=2 1x+1x33+1x55+..... =log 1+1x1-1x =log 1+(y3-1)1-(y3-1) =log y32-y3

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If $0 < y <^{2^{1 / 3}}$ and $x (y^{3} - 1) = 1$ then $\frac{2}{x} + \frac{2}{3 x^{3}} + \frac{2}{5 x^{5}} + \dots \dots \dots \dots \infty$

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$\log (\frac{y^{3}}{2 - y^{3}})$

$\log (\frac{y^{3}}{1 - y^{3}})$

$\log (\frac{2 y^{3}}{1 - y^{3}})$

$\log (\frac{y^{3}}{1 - 2 y^{3}})$

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Detailed Solution

$x (y^{3} - 1) = 1$

Given eqn

$= 2 [\frac{1}{x} + \frac{{(\frac{1}{x})}^{3}}{3} + \frac{{(\frac{1}{x})}^{5}}{5} + . . . . .] = \log (\frac{1 + \frac{1}{x}}{1 - \frac{1}{x}}) = \log (\frac{1 + (y^{3} - 1)}{1 - (y^{3} - 1)}) = \log (\frac{y^{3}}{2 - y^{3}})$

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If 0<y<21/3 and x(y3-1)=1 then 2x+23x3+25x5+…………∞

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If $0 < y <^{2^{1 / 3}}$ and $x (y^{3} - 1) = 1$ then $\frac{2}{x} + \frac{2}{3 x^{3}} + \frac{2}{5 x^{5}} + \dots \dots \dots \dots \infty$