TITLE: Compute: $\lim_{n\to\infty} \{ (\sqrt2+1)^{2n} \}$
QUESTION [7 upvotes]: 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.

REPLY [18 votes]: 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.