 Radius of convergence of a sparse power series

ID: radius-of-convergence-of-a-sparse-power-series

\sum_{n} a_{n} x^{m_{n}}

with distinct increasing integer exponents, define the ordinary coefficients to be

a_{n}

m_{n}

and zero elsewhere. The Cauchy-Hadamard theorem then gives

R^{- 1} = lim sup_{n} ∣ a_{n} ∣^{1/ m_{n}} .

The exponent in the root is the actual degree

m_{n}

, not the index

n

. For

a_{n} = (n!)^{2}

and

m_{n} = n^{2}

0 \leq 2 lo g (n!) / n^{2} \leq 2 lo g n / n \to 0

, so

R = 1

. Boundary behavior is a separate question.

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