Determining whether a series $1-\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{8}-\dfrac{1}{16}-\dfrac{1}{32}+\dfrac{1}{64}-\cdots$ converges.

Solution The series $1 - \dfrac{1}{2} - \dfrac{1}{4} + \dfrac{1}{8} -\dfrac{1}{16} - \dfrac{1}{32} + \dfrac{1}{64} - \cdots$ converges absolutely, since $1 + \dfrac{1}{2} + \dfrac{1}{4} + \cdots + \dfrac{1}{2^{n-1}}+ \cdots = \sum\limits_{k\,=\,1}^{\infty }\left( \dfrac{1}{2}\right) ^{k-1}$, a geometric series with $r=\dfrac{1}{2}$, converges. So by the Absolute Convergence Test, the series $1-\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{8}-\dfrac{1}{16}-\dfrac{1}{32}+\dfrac{1}{64}-\cdots$ converges.