A calculus proposition about sum and derivative

[tommy1729]

Hi

Let f(x) , g1(x) , g2(x) , ... be analytic on [-1,1].

If for almost Every x e [-1,1] we have :

Property A :

If
f(x) = g(x) = g1(x) + g2(x) + ... [ property I ]

And
If
f ' (x) = g1 ' (x) + g2 ' (x) + ...
[property II]

Then

f '' (x) = g1 '' (x) + g2 '' (x) + ...

And by induction the n th derivative satisfies ( n is a positive integer )

f^(n) (x) = g1^(n) (x) + g2^(n) (x) + ...

****
notice property I does not always imply property II ; example

Foerier series for x.

****

How to prove or disprove this ?

What are Nice examples ?

How about variations ? ( such as replacing first and second derivative with second and third ).

Does this motivite the desire to work with a new type of series expansions ?

Regards

Tommy1729

[tommy1729]

A variant is including property III also where this is the analogue for 3 th derivative and then making the same conclusion.

This resembles the squeezing theorem imho.

Also the q-derivate is of intrest.



Regards

Tommy1729