# 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..

• Write down the first few terms, six or so. You will spot a pattern. Prove it and conclude. Commented Dec 25, 2013 at 21:34

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

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

• You need to work with partial sums, else it is forbidden to rearrange terms (or split the infinite series in half) until absolute convergence is proven.
– zwim
Commented Jun 25, 2022 at 20:39
• You are right, so I will change my answer. However, the essential idea of my proof remains valid. Commented Jul 21, 2022 at 16:10

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 $\sum_{n=1}^\infty 1/n$.

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.

It clearly diverges

The sum is $$2/2 + 2/4 + 2/6 + 2/8 \cdots = 1+1/2+1/3 \cdots$$

• But how can i show this formally correct? Commented Dec 25, 2013 at 21:42
• induction: prove that each term will be of a certain form and simplifies to another form implying that this sum is equivalent to the harmonic sum Commented Dec 25, 2013 at 21:53