I'm trying to solve this problem from the text but I can't seem to figure out how to exactly approach it. It says,

Find the limit of the convergent sequence, $(\sqrt{2}-1)^n$ as n goes from 1 to $\infty$

I'm assuming I have to find the value of the sequence as n goes to infinity by applying the limit, but since its to the power n, I'm having bit of a trouble simplifying it. Please correct me if I'm wrong.

  • 2
    $\begingroup$ Hint: Since $1\lt\sqrt2\lt2$, $0\lt\sqrt2-1\lt1$ $\endgroup$ – robjohn Jan 15 '14 at 5:36

Note $0<\sqrt 2-1<1$. What do you know about $r^n$ if $|r|<1$?


One unique hint:

$$\lim_{n\to\infty} x^n=0\;\;\;\forall\,|x|<1$$

  • $\begingroup$ ...to rule them all and in the Darkness, bind them? (+1) $\endgroup$ – Did Jan 15 '14 at 7:40
  • $\begingroup$ "...in the Land of Mordor where the Shadows lie" . And even without a ring, in this case...:) $\endgroup$ – DonAntonio Jan 15 '14 at 10:40

One more hint:

Each number is positive and each number is less than $1/2$ the previous number, since $1 < \sqrt{2} < 1.5$.


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