Re: "minus infinity to infinity" series
- From: David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx>
- Date: Tue, 30 Aug 2005 06:43:12 -0500
On Tue, 30 Aug 2005 00:00:03 GMT, "NotP" <spam@xxxxxxxx> wrote:
>Let's say we have a function f: Z -> R (or C).
>How is convergence of the series
>sum(n = -oo, oo) f(n)
>defined?
>Are there partial sums?
>I keep seeing this notation and I don't know how to deal with it rigorously.
>I can't seem to find it in any books either.
>My guess would be something like the partial sums c_i,j = sum(n=i, j) f(n),
>and we take the limit as
>i -> -oo and j -> +oo,
There are at least two things that the notation for that sum might
mean.
One is as you suggest: The sum is the limit of what you call c_i,j as
i -> -infinity and j -> infinity.
>but I'm not quite sure how to deal with that either.
Saying that limit is L means this: For any e > 0 there exists N such
that
|L - c_i,j| < e
for all i,j satisfying i < -N and j > N.
The other thing the sum might mean is the limit of the
symmetric partial sums; that would be the limit of
c_{-i, i} as i -> infinity.
>While you're at it, what about a Riemann integral of the same form -oo to
>oo?
Same thing here. The official definition is the limit of the integral
from -B to A as A and B tend to infinity (you can figure out what
that means from what's above). There's also what's called the
"principal-value integral", which is the limit as A -> infinity
of the integral from -A to A.
>Thanks.
>
************************
David C. Ullrich
.
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