We know that corresponding to every analytic real valued function of a real variable there is a power series representation. I was just curious if the converse is true or not. The $a_n$ involved in the summation $$\sum_{n=0}^\infty a_nx^n$$ can be any random function of $n$ which could supress all possibilities of a closed form representation. If the answer happens to be true, please provide for sufficient details. Thanks in advance.
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1Your sum is just $\sum_{i=0}^n a_nx^n = (n+1)a_n x^n$. Do you mean $\sum_{n=0}^\infty a_nx^n$? – gammatester Sep 14 '17 at 14:09
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1Are you familiar with the notion of "radius of convergence"? What happens if you take $a_n = n^n$? – Mees de Vries Sep 14 '17 at 14:13
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@gammatester Edited. Thanks – NotNotLogic Sep 14 '17 at 14:28
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@MeesdeVries I think the radius of convergence would then be 0 but what are its implications? – NotNotLogic Sep 14 '17 at 14:32
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There's no associated function, then. The series would converge only for $x = 0$. (You can prove this directly.) – Mees de Vries Sep 14 '17 at 14:35
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For some coefficient sequences $a_0,a_1,a_2,\ldots$ the radius of convergence is $0.$ But when the radius is positive, then the function is analytic in the interior of the region of convergence. But that doesn't mean it has what is usually called a closed form. – Michael Hardy Sep 14 '17 at 14:51
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If the power series has a positive radius of convergence $R$ then in the interval $(-R,R)$ it defines a real analytic function. That function is unlikely to have a closed form representation if by "closed form" you mean some rational expression involving polynomials and exponential/trigonometric functions.
In fact the term by term integral of your power series will define an analytic function that is even more unlikely to have a closed form expression: see
How can you prove that a function has no closed form integral?
and
Ethan Bolker
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Small note: the guaranteed interval would be $(-R,R)$, whether the function can be extended to either bound depends on the function. – Mees de Vries Sep 14 '17 at 14:40
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In what cases will the function defined by power series in the interval $(-R,R)$ have a 'closed form' representation? Is there a criteria? – NotNotLogic Sep 14 '17 at 14:54
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By the way, yes I exactly mean that when I say 'closed form' i.e. a combination of some known functions. – NotNotLogic Sep 14 '17 at 14:55