По заданию с подробным решением.

[latex]1- \frac{1}{2}+\frac{1}{4}-\frac{1}{6}+\frac{1}{16}-\frac{1}{18}+...=\\\\= \Big (\underbrace {1+\frac{1}{4}+ \frac{1}{16}+...}_{q=\frac{1}{4}\ \textless \ 1})-\Big (\underbrace {\frac{1}{2}+\frac{1}{6}+\frac{1}{18}+...}_{q=\frac{1}{3}\ \textless \ 1}\Big )=S_1-S_2=\\\\= \frac{1}{1-\frac{1}{4}} - \frac{1/2}{1-\frac{1}{3}} = \frac{4}{3}-\frac{3}{4}= \frac{16-9}{12}=\frac{7}{12}\; ;\\\\S=\frac{b_1}{1-q}\; ,\; \; 0\ \textless \ q\ \textless \ 1 [/latex]