- #1
CantorSet
- 44
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Hi everyone,
This is not a homework question but a clarification on the following proof:
Suppose h is an infinitely differentiable real-valued function defined on /Re such that h(1/n)=0 for all n \in N. Then prove h^{(k)}(0)=0 for all k \in.
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 x_1,x_2,...,x_{k+1} as they tend to 0: h^{(k)}(0)=k! \lim_{x_1,x_2,...,x_{k+1} \rightarrow 0 } \nabla (x_1,...,x_{k+1})h
Now, choosing x_j := x_j(n) = \frac{1}{(n+j)} and letting n \in \aleph tend to infinity, we see that h^{k}(0) = 0 for all k \in N.
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?
This is not a homework question but a clarification on the following proof:
Suppose h is an infinitely differentiable real-valued function defined on /Re such that h(1/n)=0 for all n \in N. Then prove h^{(k)}(0)=0 for all k \in.
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 x_1,x_2,...,x_{k+1} as they tend to 0: h^{(k)}(0)=k! \lim_{x_1,x_2,...,x_{k+1} \rightarrow 0 } \nabla (x_1,...,x_{k+1})h
Now, choosing x_j := x_j(n) = \frac{1}{(n+j)} and letting n \in \aleph tend to infinity, we see that h^{k}(0) = 0 for all k \in N.
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?
