Question

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

Answer 1

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

Answer 2

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

Answer 3

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?

Answer 4

Sarah What about accumulation point ?

Answer 5

Sarah thx