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  4. If y=x-(x2/2)+(x3/3)-(x4/4)+ ldots and if |x|<1, then :

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Q. If $y=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\ldots$ and if $|x|<1$, then :

Bihar CECEBihar CECE 2006

Solution:

Given, $y=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\ldots$
and $|x| < 1$
Then, $y=\log _{e}(1+x)$
$\Rightarrow e^{y}=1+x \Rightarrow e^{y}-1=x$
$\Rightarrow \left(1+\frac{y}{1 !}+\frac{y^{2}}{2 !}+\frac{y^{3}}{3 !}+\ldots\right)-1=x$
$\Rightarrow \frac{y}{1 !}+\frac{y^{2}}{2 !}+\frac{y^{3}}{3 !}+\ldots=x$