Extropy
Dec16-07, 12:17 AM
Suppose that one has three numbers, v_j^0 for j=1, 2, and 3, and let their absolute values be less than or equal to 1. Suppose that one has nine continuous functions c_{jk} all of which are bounded, and thus all of which are bounded by an amount \tfrac{k}{3}.
Define
v_j^{(1)}(s)=v_j^0+\int_0^s \sum_{k=1}^3 c_{jk}(\sigma) v_k^0 d\sigma
and
v_j^{(n)}(s)=v_j^0+\int_0^s \sum_{k=1}^3 c_{jk}(\sigma) v_k^{(n-1)}(\sigma) d\sigma
It is perfectly understandable why
|v_j^{(1)}-v_j^0| \le ks,
but not so clear why
|v_j^{(n)}-v_j^{(n-1)}| \le \frac{k^n s^n}{n!}.
Somehow, one is supposed to be able to divide by n, but I do not see how is able to.
Define
v_j^{(1)}(s)=v_j^0+\int_0^s \sum_{k=1}^3 c_{jk}(\sigma) v_k^0 d\sigma
and
v_j^{(n)}(s)=v_j^0+\int_0^s \sum_{k=1}^3 c_{jk}(\sigma) v_k^{(n-1)}(\sigma) d\sigma
It is perfectly understandable why
|v_j^{(1)}-v_j^0| \le ks,
but not so clear why
|v_j^{(n)}-v_j^{(n-1)}| \le \frac{k^n s^n}{n!}.
Somehow, one is supposed to be able to divide by n, but I do not see how is able to.