3
$\begingroup$

I am trying to find the cluster point of the sequence $a_{n}:=n+(-1)^nn$. Can you please check my solution?

The subsequence diverges for increasing even $n$ since $2n$ grows infinitely.
The subsequence converges to $0$ for increasing odd $n$ since $n+(-1)n=0$

So my conclusion is that there is only one cluster point and that is $0$.

Am I fine with this?

  • 0
    "The" cluster point? Why do you think it has one, and only, one such a point? Anyway, what you did is correct.2012-12-22
  • 0
    how can i find more then ?2012-12-22
  • 0
    There are no more, but perhaps you could explain why your proof does *actually* show that zero is the only cluster point of that sequence...Check that you did so partioning the set of natural numbers in two disjoint subsets: even and odd natural numbers. This is enough and necessary to find all possible cluster point, and thus you should, perhaps, either prove it or at least mention it.2012-12-22
  • 0
    oh okay, thanks2012-12-22
  • 2
    As an aside which may or may not help, if you were computing in the extended reals rather than the reals, $+\infty$ would also be a cluster point.2012-12-22
  • 0
    Thanks Hurkyl, you are right2012-12-22

0 Answers 0