Radius of Convergence - $\sum_{n=1}^{\infty}2^n x^{n^2}$

sequences-and-series convergence-divergence power-series calculus

The radius of convergence is given by $$ \frac1{\limsup\limits_{n\to\infty}|a_n|^{1/n}} $$ In this case $$ \frac1{\limsup\limits_{n\to\infty}\left(2^n\right)^{1/n^2}}=1 $$


Apparently, one can find the radius of convergence using the root test:

$$ \limsup_{n\to\infty} |a_n|^{1/n}=\lim_{n\to\infty}|2^{-n}|^{1/n^2}=1. $$

The radius of convergence is equal to $1$.

Related

Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$
Avoiding negative integers in proof of Fundamental Theorem of Arithmetic
Let $(x_n)\downarrow 0$ and $\sum x_n\to s$. Then $(n\cdot x_n)\to 0$
How to quotient $\mathbb Z [\sqrt{-11}] / (1+\sqrt{-11})$?
Prove if a (mod n) has a multiplicative inverse, then it's unique
Solving cyclic infinite nested square roots of 2 as cosine functions
How to obtain the complete factorization of $P_n(X)=\frac{(X+i)^{2n+1} - (X-i)^{2n+1}}{2i} = 0$
If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
Evaluate $\sum_{n=1}^\infty nx^{n-1}$
Implicit differentiation
The sum $1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots-(\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\cdots)$ does not exist.
Set of branch points isn't discrete, but branch points are isolated?