My Math Forum Discontinuity of a power serie on the circle of convergence

 Complex Analysis Complex Analysis Math Forum

 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, Excuse-me my inglish is not very good ! My question:it is well-known 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]$f(z)=\sum_{k=0}^{\infty}\frac{1}{z^k}=\frac{1}{1-\frac{1}{z}}=\frac{z}{z-1}$ for |z|>1.

And be careful how you express maths,
Quote:
here you mean the unit disc with its border $|z|\leq 1$.

[/color]

 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.

 Tags circle, convergence, discontinuity, power, serie

### show that z^n does not converges open dis

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Noworry Calculus 3 May 20th, 2013 06:33 AM Breck Complex Analysis 3 October 24th, 2012 07:07 PM wawar05 Real Analysis 1 October 10th, 2012 12:07 PM rkose Number Theory 5 January 29th, 2010 07:08 AM geyikrali Real Analysis 0 April 28th, 2008 11:13 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top