April 16th, 2013, 07:26 PM  #1 
Newbie Joined: Feb 2013 Posts: 15 Thanks: 0  Strange analytic function
If r E R, we can define a very strange function fr by the following process. Express r as a decimal as r0.a1a2a3 ... where r0 is an integer and each of the ai is a digit between 0 and 9. Then the function is defined as fr(x) = r0 + infinity/E/n = 1 (an/n!) x^n Prove that this function is analytic on R for any choice of r. What kind of functions do we get if r is an integer? If r is rational? 
April 17th, 2013, 05:22 AM  #2 
Math Team Joined: Sep 2007 Posts: 2,409 Thanks: 6  Re: Strange analytic function
I am afraid you will have to explain what you mean by "infinity/E/n" for this to make any sense. Do you mean "sum for n= 1 to infinity"?

April 17th, 2013, 05:24 AM  #3 
Newbie Joined: Feb 2013 Posts: 15 Thanks: 0  Re: Strange analytic function
Yes. Sorry I guess I should have wrote it like this: infinity E n = 1 
April 17th, 2013, 12:26 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,786 Thanks: 708  Re: Strange analytic function
Function is analytic since the power series always converges, bounded by r0 + 9e^x. If r is an integer, then you have only r0. If r is rational, you have either a finite number of nonzero decimal places (polynomial) or repeated decimal (I can't describe the function).


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analytic, function, strange 
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