Determine all accumulation point of $$(a_{n})_{n \in \mathbb{N}_{+}}$$ and determine the limit (if its exists). $$a_{n}:=(-1)^{n}(1-\frac{1}{n})^{2}$$

Question

Determine all accumulation point of $$(a_{n})_{n \in \mathbb{N}_{+}}$$ and determine the limit (if  its exists).   $$a_{n}:=(-1)^{n}(1-\frac{1}{n})^{2}$$

anonymous · Answer

I get $$\lim_{n \rightarrow \infty} (-1)^{n}(1-\frac{ 1 }{ n })^{2} =\infty$$

anonymous · Answer

hmm can you prove or write in more detailed way how you got $$\infty$$ ?

anonymous · Answer

Using algebra of limits
$$\lim_{n \rightarrow \infty} (-1)^{n} \times \lim_{n \rightarrow \infty} (1-\frac{ 1 }{ n })^{2}$$
the limit of $$(1-(1/n))^{2}$$ is clearly 1

the other oscillates between 1 and -1 meaning it has no limit

so the entire sequence must diverge by the Algebra of Limits

I think that's right, maybe someone, @satellite73 @primeralph  could confirm?

anonymous · Answer

Sarah What about accumulation point ?

anonymous · Answer

Sarah thx