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April 16th, 2013, 07:26 PM   #1
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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?
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April 17th, 2013, 05:22 AM   #2
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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"?
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April 17th, 2013, 05:24 AM   #3
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Re: Strange analytic function

Yes. Sorry I guess I should have wrote it like this:

n = 1
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April 17th, 2013, 12:26 PM   #4
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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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