By recognizing the series
$$-(\frac{1}{2})-\frac{(\frac{1}{2})^2}{2}-\frac{(\frac{1}{2})^3}{3}-⋯\frac{(\frac{1}{2})^n}{n}-⋯$$
as a Taylor Series evaluated at a particular value of x, find the sum of the convergent series.
What I know so far is that this series is very similar to the series of ln(1+x), but I don't know how to make every term negative as the series ln(1+x) is an alternating series.
$$(-1)^{n-1}\frac{x^n}{n}$$
Best Answer
HINT
One standard trick is replacing $x$ by $-x$ will change in the series from $x^n$ to $(-x)^n = (-1)^n x^n$, which leaves all even numbers the same but negates all the odds.
In short, try $\ln(1-x)$ -- what series would that generate?