sequences and series - Find the limit $limlimits_{n to infty}frac{2^{-n^2}}{sumlimits_{k=n+1}^{infty} 2^{-k^2}}$

Find the limit of the following-

$$\lim\limits_{n \to \infty}\frac{2^{-n^2}}{\sum\limits_{k=n+1}^{\infty} 2^{-k^2}}$$

My work:

We can see that the denominator is in geometric series, whose sum is $1$. So taking the limit we get $\infty?$ Am I right? But I think there is a problem in this reasoning as the powers are not starting from $0$ or $1$. Can we really consider a geometric series? Any help would be great. Thanks

Answer

Hint : Multiply denominator and numerator with

$$2^{n^2}$$