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.

  • 4
    $\begingroup$ 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. $\endgroup$ – Clement C. Dec 1 '15 at 15:27
  • $\begingroup$ On the other hand, $|a_n| \le |(-1)^n|+|1/n| \le 2$ for all $n$. Also, it does not diverge. $\endgroup$ – Paolo Leonetti Dec 1 '15 at 15:36


Observe that $\displaystyle a_{2n}=1+\frac{1}{2n} \Rightarrow \lim_{ n \to \infty}a_{2n}=1$

But $\displaystyle a_{2n+1}=-1+\frac{1}{2n+1} \Rightarrow \lim_{ n\to \infty}a_{2n+1}=-1$

So limit is not unique. Thus it is not convergent.


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