I came across this (cool) question this weekend
Find the limit of the following sequence as $n$ approaches infinity.
$x_1 = 1$ and $x_{n+1} = \sqrt{x_n^2+\frac{1}{2}^n}$
I had two questions about it. I approximated it using excel to be about 1.224745. Does anyone know if there is an exact expression for what this converges to? Also, on problems like this, I normally use the fact that if a sequence converges all of its sub-sequences converge, and have the same limit. Plugging in $L$ for all of the $x_n$'s and solving didn't get me anywhere. Why does this "trick" not work here, and what other sort of recursive sequences does it fail for? What are some other methods for finding the limit in a situation like this? Thanks in advance.
Best Answer
For this particular sequence, it is easier to deal with $x_n^2$ as
$$x_{n+1}^2 = x_n^2 + \frac{1}{2^n}.$$
Thus
$$\lim_{n\to \infty} x_n^2 = \left(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} +\cdots \right)= 2$$
Thus
$$\lim_{n\to \infty} x_n = \sqrt{\lim_{n\to \infty} x_n^2}=\sqrt{2}$$
which is not the same as what you've got. Your number looks like $\sqrt{1.5}$ to me.