Convergence of $\sum\limits_{n=1}^{\infty}\frac{1+(-1)^n}{n}$ I want to check, whether $\sum\limits_{n=1}^{\infty}\frac{1+(-1)^n}{n}$ converges or diverges.
Leibniz's test failed, and ratio test just made it even more complicated, so i tried to use the comparison test, but i can't find a suitable series so that $\lim\limits_{n \rightarrow \infty} \frac{a_n}{b_n}$ exists..
 A: To prove: $\sum_{n\geq 1} \frac{1+(-1)^n}{n}$ diverges.
Proof: \begin{align*} \sum _{n\geq 1} \frac{1+(-1)^n}{n} &= \sum _{k\geq 1} \frac{1+(-1)^{2k}}{2k} + \sum _{k\geq 1} \frac{1+(-1)^{2k-1}}{2k-1} \\
&= \sum _{k\geq 1} \frac{2}{2k} + \sum _{k\geq 1} \frac{0}{2k-1} \\\
&= \sum _{k\geq 1} \frac{1}{k} \end{align*}
Because $\sum _{k\geq 1} \frac{1}{k}$ diverges, $\sum_{n\geq 1} \frac{1+(-1)^n}{n}$ diverges as well. \qed
Improved answer (with partial sums):
For each $m\in \mathbb{N}$ we have
\begin{align*}
\sum _{n=1} ^m \frac{1 + (-1)^n}{n} 
& = \sum _{k=1} ^{\lfloor m/2 \rfloor} \frac{1 + (-1)^{2k}}{2k} + \sum _{k=1} ^{\lfloor \frac{m+1}{2}\rfloor} \frac{1 + (-1)^{2k-1}}{2k-1} \\
& = \sum _{k=1} ^{\lfloor m/2 \rfloor} \frac{2}{2k} + \sum _{k=1} ^{\lfloor \frac{m+1}{2}\rfloor} \frac{0}{2k-1} 
= \sum _{k=1} ^{\lfloor m/2 \rfloor} \frac{1}{k}
\end{align*}
Because $\sum _k 1/k $ diverges, $\sum _n \frac{1 + (-1)^n}{n}$ diverges as well. QED
A: Hint: The partial sums are $$S_N=H_{\lfloor N/2 \rfloor} $$ where $H_k$ is the sequence of harmonic numbers, i.e. the partial sums of the (divergent) harmonic series $\sum_{n=1}^\infty 1/n$.
A: Since the summation is just an addition of the sequence $a_n=\frac{1+(-1)^n}{n}$, let's examine the sequence $a_n$. 
For $n$ odd, we have $a_{\text{odd}}=\frac{1+-1}{n_{odd}}=0$. So the only terms that remain are the even terms which are $a_{even}=\frac{1+1}{n_{even}}=\frac{2}{n_{even}}$. Since $n$ is even, here it is of the form $n=2k$ for $k \in \mathbb{Z}_+$. So the terms of our series which are nonzero are 
$$
a_{even}=\frac{2}{n_{even}}=\frac{2}{2k}=\frac{1}{k}
$$
If we add all these we get
$$
\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots
$$
which is the harmonic series, which clearly diverges. 
A: It clearly diverges
The sum is
$$
2/2 + 2/4 + 2/6 + 2/8 \cdots = 1+1/2+1/3 \cdots
$$
