Q.The value of $1 - \dfrac{1}{2}\left(\dfrac{2}{3}\right) + \dfrac{1}{3}\left(\dfrac{2}{3}\right)^2 - \dfrac{1}{4}\left(\dfrac{2}{3}\right)^3 + \ldots$ is:

Logarithmic Series

The logarithmic series is the power-series (Maclaurin) expansion of the natural logarithm $\log(1+x)$. It states that

and this expansion is valid for $-1 < x \le 1$. Replacing $x$ by $-x$ gives the companion series $\log(1-x) = -\left(x + \dfrac{x^2}{2} + \dfrac{x^3}{3} + \cdots\right)$, valid for $-1 \le x < 1$. Here $\log$ denotes the natural logarithm to base $e$.

These expansions let us sum many related series and estimate logarithms. Subtracting the two gives a rapidly useful form:

$\log\!\left(\frac{1+x}{1-x}\right) = 2\left(x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots\right).$ …