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April 4th, 2013, 08:57 AM  #1 
Newbie Joined: Apr 2013 Posts: 3 Thanks: 0  Discontinuity of a power serie on the circle of convergence Good evening to all members, Excuseme my inglish is not very good ! My question:it is wellknown that if f(z) is the sum of a power serie "Sum of a_n z^n " of radius 1 which converges for z=1(that is to say "Sum of a_n converges"), then the fonction f is continue in every triangle inclused in the open disk of convergence B(0,1) and which has 1 as a vertex( it comes from the uniform convergence of the serie in this triangle). I would like to find such a power serie such as the function f is not continue in the open disk of radius 1 U{1}= B(0,1)U {1}. ( obviously it is seems sufficent to find a sequence (z_n) which converges towards 1 "interiously tangently" in the B(0,1) and such that f(z_n) does not converges towards f(1)). Thank you to everybody who will take time to answer tome Have a good day Triple 0. 
April 4th, 2013, 11:14 AM  #2  
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Discontinuity of a power serie on the circle of converge [color=#000000] for z>1. And be careful how you express maths, Quote:
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April 5th, 2013, 11:13 PM  #3 
Newbie Joined: Apr 2013 Posts: 3 Thanks: 0  Re: Discontinuity of a power serie on the circle of converge
Thank you very mutch ZardoZ, but the exemple you give me is not a power series of radius 1 (that is to say a function of the kind "Sum of a_n z^n) and your fonction does not exist for z= 1 . Apart of that, for me " B(0,1) U {1}" means the union of the open ball of radius 1 and the single point 1 alone. Have a good evening and thank you or your help, Triple 0. 
April 6th, 2013, 12:59 AM  #4 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Discontinuity of a power serie on the circle of converge [color=#000000]There are no balls on the complex plane. Anyway the context of your question is confusing! [/color] 
April 6th, 2013, 12:52 PM  #5 
Newbie Joined: Apr 2013 Posts: 3 Thanks: 0  Re: Discontinuity of a power serie on the circle of converge
Thank you Zardoz, excuse me i said "ball" but i should have say unit open disk in the complex plane: By B(0,1) i mean the unit disk in the complex plane that is to say B(0,1) = {z belonging C / IzI < 1} Good morning to all of you, Triple 0. 

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