# Find if the sequence $\left((-1)^n + \frac1n\right)_{n\geq 1}$ converges or diverges

I am trying to find out if the following sequence $(a_n)_{n\geq 1}$ will either converge or diverge. $$a_n = (-1)^n + \frac1n$$

If the limit converges, I need to find its limit.

I tried plugging in various numbers for n besides 1, but I am not entirely sure if it really does converge.

• Consider the two subsequences $(a_{2n})_n$ and $(a_{2n+1})_n$. If the sequence $(a_n)_n$ converges, then these two must also converge to the same limit. – Clement C. Dec 1 '15 at 15:27
• On the other hand, $|a_n| \le |(-1)^n|+|1/n| \le 2$ for all $n$. Also, it does not diverge. – Paolo Leonetti Dec 1 '15 at 15:36

$a_n=(-1)^n+\frac{1}{n}$
Observe that $\displaystyle a_{2n}=1+\frac{1}{2n} \Rightarrow \lim_{ n \to \infty}a_{2n}=1$
But $\displaystyle a_{2n+1}=-1+\frac{1}{2n+1} \Rightarrow \lim_{ n\to \infty}a_{2n+1}=-1$