Consider a function that can be represented by its Taylor expansion in $a$
$$ f(x) = f(a) + \sum_{k}^{\infty} \frac{f^{(k)}(a)}{ k!} (x-a)^k$$
Now for a fixed a, the points where $f(x) = f(a)$ are exactly those points where the sequences $(\frac{f^{(k)}(a)}{ k!})_k$ and $((x-a)^k)_k$ are "orthogonal", although they may not be in a Hilbert space.
My questions are:
Is this notion of orthogonality "useful" in the sense that it provides some additional insight on Taylor series?
Can one explicitly derive $$(\frac{f^{(k)}(a)}{ k!})_{k}^{T} = \{(b_k) | \sum_{k}^{\infty} \frac{f^{(k)}(a)}{ k!} b_k = 0 \}?$$
What about for a fixed $(b_n)$ determining $a$ such that $\sum_{k}^{\infty} \frac{f^{(k)}(a)}{ k!} b_k = 0$ ?