for this problem i think i need to use the abel's lemme which states that

for r,r_0 in the reals such that 0<r<r_0, if there exists real number M

such that M>0 and (abs(a_n))(r_0)^n≤M for all natural numbers, then

the series Σa_nz^n for n≥0 converges normally for abs (z)<r.

converging normally means that if (u_n(z))_n is a sequence of complex

valued functiosn on E, then the series Σu_n converges normally on E

given that the series Σllull converges (llull is the sup-norm of u,

defined by llull=sup(z in E) abs (u(z))

problem: if ρ is the radius of convergence of the series Σa_nz^n

(n≥0), and r is a real number less than ρ, prove that the

latter series converges normally for abs (z)<r (btw z is a modulus defined on each complex number by abs (z) = sqrt (x^2+y^2) and with property abs (z_1*z_2)=abs(z_1)abs(z_2) and abs (z_1+z_2)<= sqrt (z_1)+sqrt (z_2)