# An analogue of the Cauchy formula for radius of convergence for power series with arbitrary (non-integer) exponents

By Cauchy formula, the radius of convergence of the series $\sum_{n=0}^{\infty}a_nr^{n}$ is $\rho=1/\limsup\limits_{n\rightarrow +\infty}\sqrt[n]{|a_n|}$.

Let $\{\lambda_n\}_{n=0}^{\infty}$ be an arbitrary sequence of non-negative increasing real numbers that tend to $+\infty$. Consider the power series $\sum_{n=0}^{\infty}a_nr^{\lambda_n}$. Does there exist certain conditions on the coefficients $a_n$ and on the exponents $\lambda_n$ under which the above series converge in some neighborhood of 0 and is there an analogue of the above formula for the radius of convergence for such series?