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  4. The value of 1- log 2+(( log 2)2/2!)-(( log )3/3!)+... is

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Q. The value of $ 1-\log 2+\frac{{{(\log 2)}^{2}}}{2!}-\frac{{{(\log )}^{3}}}{3!}+... $ is

Jharkhand CECEJharkhand CECE 2008

Solution:

In a given series there is a factorial in denominator, therefore it may be in the form of exponential series.
$ \therefore $ $ 1-\log 2+\frac{{{(\log 2)}^{2}}}{2!}-\frac{{{(\log 2)}^{3}}}{3!}+... $
$ ={{e}^{-\log 2}} $
$ ={{e}^{\log {{2}^{-1}}}} $
$ ={{2}^{-1}}=\frac{1}{2} $