7
$\begingroup$

Compute the following limit:

$$\lim_{n\to\infty} \{ (\sqrt2+1)^{2n} \}$$ where $\{x\}$ is the fractional part of $x$. I need some hints here. Thanks.

  • 0
    I think $\{(\sqrt2+1)^{2n}\}=2^n\sqrt2$2012-06-22

1 Answers 1

18

Consider $$ (\sqrt2+1)^{2n} + (\sqrt2-1)^{2n} $$

Try to show that it is an integer and hence this fractional part you are looking for is $1 - (\sqrt2-1)^{2n}$ Now the limit becomes easy.

  • 0
    got it! Thanks!2012-06-22
  • 0
    It's limit is coming out to be 1, but fractional part can't be 1, so the result is 0 or 1?2012-06-22
  • 1
    I don't see why the fact that no element in the sequence can be 1 should stop the limit from being 1.2012-06-22
  • 1
    okay!, i got your point :).Thanks!!2012-06-22