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[tex] f(x) = x^5ln(x) [/tex]

I can differentiate [tex]f(x)[/tex] say 6 times and I'm left with just [tex]ln(x)[/tex]. But this is not a proof!

How do I go about proving it?

Could I use the Taylor expansion?

- Thread starter j-lee00
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- #1

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[tex] f(x) = x^5ln(x) [/tex]

I can differentiate [tex]f(x)[/tex] say 6 times and I'm left with just [tex]ln(x)[/tex]. But this is not a proof!

How do I go about proving it?

Could I use the Taylor expansion?

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Office_Shredder

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Say [tex]f(x)=x^5[/tex] and [tex]g(x)=ln(x)[/tex] but f(x) is not infinitely differentiable ?

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So if the derivative = 0, then it is still differentiable?

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Yes. To see this, just apply the definition of derivative to f(x) = 0.

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statdad

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If you differentiate [tex] f[/tex] 6 times you are not left with [tex] \ln(x) [/tex].

[tex] f(x) = x^5ln(x) [/tex]

I can differentiate [tex]f(x)[/tex] say 6 times and I'm left with just [tex]ln(x)[/tex]. But this is not a proof!

How do I go about proving it?

Could I use the Taylor expansion?

- #11

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Simply out of curousity.. Where can i find the proof for the uniqueness of taylor series of function?^{5}*ln(x) is that ln(x) doesn't have a taylor series expansion about 0.

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Landau

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Well, there's not really anything to prove: the Taylor series isSimply out of curousity.. Where can i find the proof for the uniqueness of taylor series of function?

Interesting (in complex analysis): two convergent power series which coincide on an infinite set with 0 as accumulation point, must be equal. See e.g. Serge Lang's book, p.62.

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Office_Shredder

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1. Every polynomial of degree n can be differentiated (n+1) times before vanishing (it is n+1 times differentiable).

2. Thus, an infinite order polynomial is infinitely differentiable.

3. The power series expansion of ln x is of infinite degree. This expansion absorbs the x^5 term, merely creating another infinite degree expansion with each term 5 degrees higher. This combined expansion is infinitely differentiable.

I don't know what rigor my proof lacks, but that's my take if it's at all valuable.

NVM: I ignored radius of convergence as pointed out by the previous poster. HOWEVER, can one use analytic continuation to "cover" the entire domain of the function?

- #15

statdad

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A polynomial is infinitely differentiable - every derivative after the n+1st is zero, but that is immaterial.Here's what I would do:

1. Every polynomial of degree n can be differentiated (n+1) times before vanishing (it is n+1 times differentiable).

There is no such thing as an infinite order polynomial.2. Thus, an infinite order polynomial is infinitely differentiable.

This is correct heuristically, and may be all the OP needs. But I would expect that the statements made in (1) and (2) would still run afoul of his/her professor's grading. The correct terms must be used. Even so, it is possible to show that the function is infinitely differentiable without resorting to power series.3. The power series expansion of ln x is of infinite degree. This expansion absorbs the x^5 term, merely creating another infinite degree expansion with each term 5 degrees higher. This combined expansion is infinitely differentiable.

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Office_Shredder

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