My Math Forum Strange analytic function

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 April 16th, 2013, 07:26 PM #1 Newbie   Joined: Feb 2013 Posts: 15 Thanks: 0 Strange analytic function If r E R, we can de fine 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: 6,834 Thanks: 733 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 non-zero decimal places (polynomial) or repeated decimal (I can't describe the function).

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