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This is not a homework question but a clarification on the following proof:

Suppose [itex]h[/itex] is an infinitely differentiable real-valued function defined on [itex]/Re[/itex] such that [itex]h(1/n)=0[/itex] for all [itex] n \in N [/itex]. Then prove [itex]h^{(k)}(0)=0[/itex] for all [itex] k \in [/itex].

Proof: Since h is infinitely differentiable in a neighborhood of 0, the kth derivative of h at 0 is the limit of its normalized (k+1)th divided difference at distinct nodes [itex]x_1,x_2,...,x_{k+1}[/itex] as they tend to 0: [tex] h^{(k)}(0)=k! \lim_{x_1,x_2,...,x_{k+1} \rightarrow 0 } \nabla (x_1,...,x_{k+1})h [/tex]

Now, choosing [itex] x_j := x_j(n) = \frac{1}{(n+j)}[/itex] and letting [itex] n \in \aleph [/itex] tend to infinity, we see that [itex] h^{k}(0) = 0 [/itex] for all [itex] k \in N [/itex].

I don't understand the proof above since I don't know what normalized (k+1)th divided difference at distinct nodes means. Does anyone know?

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# What does normalized (k + 1)th divided difference at distinct nodes mean?

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