Methods of finding the difference to the nearest integer? The question asked to find the "smallest value of n such that $(1+2^{0.5})^n$ is within 10^-9 of a whole number." I'm unsure of the approach to the question.
The question was in the chapter of 'binomial expansion' in the textbook.
Thanks for your time!

For such problems, the concept of conjugated terms is important.
If you look at the binomial expansion of $(1+\sqrt{2})^n$ and $(1-\sqrt{2})^n$ you will see that all the terms including $\sqrt{2}$ in an odd power will have opposite signs. Hence
$$(1+\sqrt{2})^n+(1-\sqrt{2})^n$$ is always an integer.
But fortunately $|1-\sqrt{2}| \approx 0.4<1$ and so the second term quickly converges to 0 which makes $(1+\sqrt{2})^n$ getting closer to an integer value.
Now, to estimate how close this is you have to analyse how quick $(1-\sqrt{2})^n$ converges to 0. Can you continue from here?
