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Radius of convergence after powering coefficients

If

L = lim sup a_{n}^{1/ n}

for positive coefficients, then the coefficients

a_{n}^{2}

have root limsup

L^{2}

. Coefficients

a_{n}^{a_{n}}

instead have root terms

a_{n}^{a_{n} / n} = exp (\frac{a _{n} l o g a _{n}}{n}),

so their radius depends on the growth of

a_{n} lo g a_{n}

relative to

n

.

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