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advanced sum problem ?
#1
what is the radius of convergence for

sum ( r over all rationals >= 0 ) a_r z^r for real a_r coefficients and when expanded at z = a ?

a_r is analytic.

in fact how to change the expanding point ? so lets start the question for z= a = 0.

i assume for bounded a_r integral a_r dr must have oo zero's on the real line is a necc condition.

maybe for bounded a_r and integral a_r z^r dr converges and has oo zero's on the real line for real z determines the radius >= z ( expanded at 0 ).
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