The value of $log (10 + 10 x + 10 x^{2} + . . . . . .)$

1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + . . . . .

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1 - x + \frac{x^{2}}{2!} - \frac{x^{3}}{3!} + . . . .

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1 - x - \frac{x^{2}}{2} - \frac{x^{3}}{3} + . . . .

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1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + . . . . .

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Solution

The correct option is B $1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + . . . . .$
$log 10 (1 + x + x^{2} + . . .) = log 10 \times \frac{1}{1 - x} = log 10 + log \frac{1}{1 - x}$
$= 1 - log (1 - x) = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{3} + . . .$

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