Let's consider the series: $\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}(-1)^n$.
I would suggest that it doesn't converge. One can see it as follows:
Let be $S_{2n}:=\sum\limits_{k=1}^{2n}\frac{(-1)^k}{k}(-1)^{2n}$ and $S_{2n+1}:=\sum\limits_{k=1}^{2n+1}\frac{(-1)^k}{k}(-1)^{2n+1}$ two subsequences (subseries?) of $\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}(-1)^n$. Both converge due to alternating series test (Leibniz criterion). However, if I take a look at $|S_{2n}-S_{2n+1}|$, I notice:
$$|S_{2n}-S_{2n+1}|=\Big|\sum\limits_{k=1}^{2n}\frac{(-1)^k}{k}(-1)^{2n} -\sum\limits_{k=1}^{2n+1}\frac{(-1)^k}{k}(-1)^{2n+1}\Big|\\=\Big|2\sum\limits_{k=1}^{2n}\frac{(-1)^k}{k}(-1)^{2n} -\frac{1}{2n+1}\Big|=2\sum\limits_{k=1}^{2n}\frac{(-1)^{k-1}}{k} +\frac{1}{2n+1}>1.
$$
Hence, $\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}(-1)^n$ violates the Cauchy-criterion, or in other words I can always find two partial sums which are not arbitrarily close to each other.
Is this correct? Is there a faster or more elegant way two show this result?
Best Answer
The sequence $$u_n =\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}$$
converges to $-\ln(2) < 0$. Therefore the sequence $$\sum\limits_{k=1}^{n}\frac{(-1)^k}{k}(-1)^n=(-1)^n u_n$$
does not converge.