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If $\frac{1}{(1 - 2 x) (1 + 3 x)}$ is to be expanded as a power series of x, then

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$- 1 / 3 < x < 1 / 2$

$| x | < 1 / 6$

$| x | < 1 / 2$

$| x | < 1 / 3$

answer is D.

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Detailed Solution

$\frac{1}{(1 - 2 x) (1 + 3 x)} = \frac{1}{5} (\frac{2}{1 - 2 x} + \frac{3}{1 + 3 x}) \Rightarrow \frac{1}{(1 - 2 x) (1 + 3 x)} = \frac{2}{5} {(1 - 2 x)}^{- 1} + \frac{3}{5} {(1 + 3 x)}^{- 1} is valid if | 2 x | < 1 and | 3 x | < 1$

$|x| < \frac{1}{2}, |x| < \frac{1}{3} \Rightarrow |x| < \frac{1}{3}$

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