Asymptotic Power Series Behavior

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Homework Statement

Suppose [tex]f(x) = \sum_1^\infty a_n x^n [/tex] is a power series such that [tex] \lim a_{n+1}/a_n \to 1/n [/tex]. Show that the magnitude of f(x) grows asymptotically as [tex] e^x [/tex].

This is not a homework question. But, if I know why it is true (or if it is), then I can use it to answer a quantum mechanics question.

The Attempt at a Solution

If the coefficients were actually in ratios of 1/n, as opposed to limiting to that ratio, then after factoring out the first coefficient, you would be left with the power series for [tex] e^x [/tex].

Or, one can easily show that f(x) grows faster than any polynomial, but that doesn't quite get us there either.

f(x) has infinite radius of convergence, so it converges uniformly everywhere.

The difficulty is that there are three limits. You want the limit of the coefficients to determine the limit of the function which is the limit of the series of those coefficients.

Any help would be appreciated. Thanks in advance.

Clarification
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By [tex] \lim a_{n+1}/a_n \to 1/n [/tex], I meant [tex] a_{n+1}/a_n [/tex] grows theta of [tex] 1/n [/tex].
I'm trying to translate my physics notes into something mathematical. There, it just says, "for large n [tex] a_{n+1}/a_n [/tex] goes like [tex] 1/n [/tex], which is the same as the Taylor series for [tex] e^x [/tex], so the power series of [tex] a_n [/tex] has the same asymptotics as [tex] e^x [/tex]."
I feel like there's some analysis which validates this claim, but it's hard to translate into precise language.

 

[eczeno]

Homework Statement

Suppose [tex]f(x) = \sum_1^\infty a_n x^n [/tex] is a power series such that [tex] \lim a_{n+1}/a_n \to 1/n [/tex]. Show that the magnitude of f(x) grows asymptotically as [tex] e^x [/tex].

it is not clear what you mean by [itex] \lim a_{n+1}/a_n \to 1/n [/itex]. if it is the limit as n approaches infinity (or as n approaches anything), then the result cannot have n in the expression. if the ratio [itex]a_{n+1}/a_n[/itex] is always of the order [itex]1/n[/itex] then that shows the radius of convergence is infinite. however, i am not sure what it says about the growth of the power series. certainly the coefficients for [itex]e^x[/itex] have this pattern, so I think the idea that any series with this pattern will grow exponentially is probably correct. However, I don't really have an idea as to how to show that.

cheers.