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magic_box

Guest

taking 1/2^nth derivative infinite times?

I have been doing some more thinking, and I was wondering whether or not this is true;

If the series

,

then for any function

,

will taking the 1/2 derivative, then the 1/4 derivative, then the 1/8 derivative, then the 1/16 derivative (and so on) yield the 1st derivative, since the series converges? Just a thought.

EDIT: I tried writing this out as an infinite series; you end up with the terms x^(1/2), x^(3/4), x^(7/8) ...

. Since you are multiplying these terms, you end up with

since the series diverges... so does that mean that taking the (1/2^n)th derivative repeatedly (infinite times) yields infinity? Strange... I would have thought it to be 1, because of the convergence of the summation of 1/2^n from n = 1 to infinity.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: taking 1/2^nth derivative infinite times?

Hi;

I am not following what you are trying to do. If we choose f(x) = x and take successive derivatives of 1 / 2,  1 / 4, 1 / 8, 1 / 16 + ... + How are you figuring what that is going to equal?

We need a value for x do we not?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

aleclarsen12

Member

Registered: 2008-06-01

Posts: 36

Re: taking 1/2^nth derivative infinite times?

will taking the 1/2 derivative, then the 1/4 derivative, then the 1/8 derivative, then the 1/16 derivative (and so on) yield the 1st derivative, since the series converges? Just a thought.

Correct me if I'm wrong but I do not believe there is a such thing as taking a fractional derivative.

That being said, assuming there is, I believe it would have to work something like this:

What your saying makes sense theoretically but I do not understand the idea of a "1/2 derivative". :\ Are there also "1/2 integrals"?

EDIT: I did some Googling; apparently there is a "half derivative". Sorry to question to concept.

Last edited by aleclarsen12 (2011-01-02 06:53:04)

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bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: taking 1/2^nth derivative infinite times?

Hi;

Yes, there is such a thing as a fractional derivative and a fractional integral.Not only 1 / 2.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

zetafunc.

Guest

Re: taking 1/2^nth derivative infinite times?

This looks like an interesting post... is this possible? I know about fractional derivatives but would that converge to the first derivative?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: taking 1/2^nth derivative infinite times?

Hi zetafunc.;

I do not know much about fractional derivatives or integrals
except that they can be calculated.

If you just want to do research you can google for them.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.