TITLE: $x_n$ such that $x_n$ does not converge but $2(x_n)^n$ does to a non-zero real number.
QUESTION [0 upvotes]: Find a sequence $x_n$ such that $x_n$ does not converge but $2(x_n)^n$ does to a non-zero real number.
I have already found such a sequence that converges to zero, is there a sequence that converges to a non-zero real number?

REPLY [1 votes]: Consider $x_n = (-1)^{n + 1}$. Then $x_n^n = (-1)^{n (n + 1)}$. And since either $n$ or $n + 1$ is even, $n(n + 1)$ is always even, so $x_n^n = 1$ for all $n$.
If you restrict $x_n$ to be positive then there are no such examples. We can show that if $\lim\limits_{n \to \infty} x_n^n$ is nonzero and the $x_n$ are all positive then $\lim\limits_{n \to \infty} x_n = 1$.
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