EXAMPLE $1$ The alternating series

$\frac{1}{1}\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+$dots$={\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^{n-1}}{n}$

satisfies

$($i$)b_{n+1}?vv{b}_n$ because $\frac{1}{(n+1)}<mfrac\; id="EditorElement\_134584"><mrow\; id="EditorElement\_134577"><mn\; id="EditorElement\_134574">1</mn></mrow><mrow\; id="EditorElement\_134583"><mi\; id="EditorElement\_134580">n</mi></mrow></mfrac>$

$($ii$)\underset{n}{lim}{b}_n=\underset{n}{lim}\frac{1}{n}=$

so the series is convergent by the Alternating Series Test.