For any polynomial function f(x) if you know f(a), f'(a), f"(a), f"'(a)... for some value of a you can reconstruct the function. This is true even if the polynomial has an infinite number of terms.

(f' is the first derivative of f, f" is the second derivative etc.)

Define s(a) to be the sequence of f(a), f'(a), f"(a), f"'(a), ...

For each of the following, find the function:

(1) s(1) = 19, 23, 32, 18, 0, 0, 0, 0, ...

(2) s(0) = 1, 2, 4, 8, 16, 32, ...

(3) s(1) = 1, -1, 1/2, -1/6, 1/24, -1/120, 1/720, ...

(4) s(0) = 0, 1, 0, -1, 0, 1, 0, -1, ...

(5) s(0) = ln(2), ln(2^ln(2)), ln(2^{ln(2ln(2) )}), ...

(In reply to Partially done but help needed (Spoiler) by Harry)

1 and 2 are correct.
I don't understand what you are trying for #3 but it should be a simple function.
#4 is almost right but sinh doesn't alternate signs so it must be some other.
#5 is not quite right.

Posted by Jer on 2009-04-02 13:35:30