Leibniz's Alternating Series Test
The series $\sum (-1)^{k-1} u_k$ converges if:
- $u_k \geq 0$
- $u_{k+1} \leq u_k$
- $u_k\rightarrow0$ as $k\rightarrow\infty$
I need to find an alternating series which diverges because it fails to satisfy the second condition (i.e. $u_{k+1} > u_k$). Can anybody help me find such a series?
Best Answer
You can take the series
$\displaystyle\sum_{k=1}^{\infty}(-1)^{k-1}\frac{\frac{1}{2}(3+(-1)^{k-1})}{\frac{1}{4}(2k+1+(-1)^{k-1})}=\frac{2}{1}-\frac{1}{1}+\frac{2}{2}-\frac{1}{2}+\frac{2}{3}-\frac{1}{3}+\frac{2}{4}-\frac{1}{4}+\cdots$
(Notice that $\dfrac{1}{n}<\dfrac{2}{n+1}$ for $n>1$, and
that the even partial sums of the series are the partial sums of the harmonic series.)