Re: incresing functions on [0, inf)

From: Amanda (sca18_at_hotmail.com)
Date: 08/25/04 

Date: 25 Aug 2004 09:06:21 -0700

I'm not sure what all that is about. Is there an easily stated problem
> where this originates?

This is related to the planning of an electrical system. I'm trying to
find a sequence that converges to the marginal cost.

More specfically:

I have a sequence (C_n) that converges on [0, inf) to a function C,
which expresses the total cost of supplying a load x. For evey n, C_n
is differentiable and there are functions f_n and g_n such that C'_n =
f_n + g_n on [0,inf]. The f_n's are strictly increasing on [0, inf)
and f_n converges on [0, inf) to a continuous function f. For every n,
g_n(x) -> 0 as x-> inf and g_n converges on [0, inf) to the
identically zero function.
Based on these conditions, we can affirm the convergence f_n -> f is
uniform on [0,M] for every M>0. And it's immediate that f_n+ g_n
converges to f. But I'm not sure if, as desired, f = C'.
If I could assure g_n converges to 0 uniformly on [0, M] for every
M>0, then it'd be true that f_n + g_n converges uniformly to f on [0,
M] and we'd actually have that f = C' on [0,M] for every M, so f = C'
on the whole [0, inf).
But the problem is, based on such information, is it possible to
affirm g_n -> 0 uniformly on [0, M] for every M>0?
If the g_n's are continuous and g_n is a monotonic sequence, then I
could apply Dini's theorem, right?
Amanda

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