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Say we have $$\sum_{n\ge0}a_nx^n$$

Where$\:\:a_n=\begin{cases} \Phi_n, & \text{if}\ \:n\:\:\text{even} \\ \Theta_n, & \text{if}\:\:n\:\:\text{odd} \end{cases}$

Now how do we go about finding the set of values of$\:x\:$for which the series converges?

Do we add each convergence radius if such a thing is doable?

Or, if the radii are such that$\:\underset{even}{\mathbf{R}}\subset \underset{odd}{\mathbf{R}}\:$, do we simple choose the most restrictive of the latter two?

Thanks!

User 123732
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  • The radius of convergence is defined as $$\rho=\limsup \sqrt[-n]{|a_n|}$$ then, if we know the structure of odd and even coefficients we can see what is this value. – Masacroso Oct 31 '16 at 18:23

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