# How to find Limit points for $A = \left( (-1)^{m} + \dfrac{(-1)^n}{n}\right)$

I need to find the limit point for the following set

$$A = \left( (-1)^{m} + \dfrac{(-1)^n}{n}\right) \text{m,n} \in \mathbb N$$

So, Here is what I did , If I fix $$m$$ and vary $$n$$ then as $$n \to \infty$$ clearly $$+1,-1$$ are two limit points.

Also If I fix $$n$$ and vary $$m$$ then $$\left( 1 + \dfrac{(-1)^n}{n}\right)$$ and $$\left( -1 + \dfrac{(-1)^n}{n}\right)$$ are other two limit points.

Is this correct ?

## 1 Answer

If we refer to $$m,n\to \infty$$ then since $$\dfrac{(-1)^n}{n}\to 0$$ all boils down in $$(-1)^{m}$$ then limit points are $$1$$ and $$-1$$, otherwise also the other two limit points exists and they are dependent upon $$n$$.

• But In the second case I have fixed a value $n \in N$. Then why are you taking taking the limit $n \to \infty$ ? Can you please explain this. Oct 25, 2019 at 17:32
• @zeroflank If we refer to $m,n\to \infty$ then limit points are just $\pm1$ otherwise yes you are correct there are also other two limit points depending upon $n$.
– user
Oct 25, 2019 at 17:36
• Thank you, for such a quick response. Oct 25, 2019 at 17:41