Understanding Power Series Derivatives: A Question

[fresnelspot]

Hi


My confusion is about this power series.

If derivative of a function (f(x)) is another function(g(x)) then, this holds for the series terms of the functions.
My question is If one knows this derivation relation Just two equal labelled series terms of two other functions. And one term is exactly the differential of the other Then can we generelize this relation for the "big " functions?
Thanx

 

[mathman]

Your message is confusing. Among other things what are "big" functions? Also what do mean by "equally labelled series"?

 

[fresnelspot]

:)

by "big " functions i mean the sum of all terms .
by "equal labelled" i mean they both will be the nth terms of the series expansions they belong.

 

[Defennder]

Let me see if I understand this. You're saying that a function f(x) is a derivative of another function g(x) if the nth term of f(x) is a derivative of the (n+1)th term of the power series of g(x)? Or are you asking if the converse is true? Or something else entirely?

 

[rbj]

Let me see if I understand this. You're saying that a function f(x) is a derivative of another function g(x) if the nth term of f(x) is a derivative of the (n+1)th term of the power series of g(x)? Or are you asking if the converse is true? Or something else entirely?

just to be clear with Defennder's question, if

$$f(x) = \sum_{n=0}^{+\infty} a_n x^n$$

and

$$g(x) = \sum_{n=0}^{+\infty} b_n x^n$$

and $f(x) = (d/dx)g(x)$

then we know that

$$a_n = (n+1)b_{n+1} \ \ \ \forall n \ge 0$$

knowing all of the a_n is sufficient to tell us what the b_n are except for b₀, which is information that is lost forever when g(x) is differentiated.

dunno if that is the question that was asked.

 

[fresnelspot]

I am very sorry for my verbal skills .The problem is confusing for me so it becomes more difficult to explain it in words.
My problem is this :
The concept is clear when we see the relation between f and g and reflect this relation to the series terms.

What if we know there is a derivation relation between two terms ( we only know these terms . For example we have experient data of these two functions for a limited region . they are both kth series terms of their function ) . Can we generalize this derivation relation to the functions

 

[rbj]

My problem is this :
The concept is clear when we see the relation between f and g and reflect this relation to the series terms.

What if we know there is a derivation [derivative?] relation between two terms ( we only know these terms . For example we have experient data of these two functions for a limited region . they are both kth series terms of their function ) . Can we generalize this derivation relation to the functions

no, only knowing that

$$a_n = (n+1)b_{n+1}$$

for some particular n is not sufficient to say that $f(x) = (d/dx)g(x)$. you must know that

$$a_n = (n+1)b_{n+1} \ \ \ \forall \ n \ge 0$$

which means for all $\ n \ge 0 \$ to be able to conclude that $f(x) = (d/dx)g(x)$.

 

[fresnelspot]

thanx