/sci/ - Science & Math

>>7096991
Yes and no. Arithmetic doesn't allow to compute any infinite sum.
If you say
"Call <span class="math">\sum_{n=1}^\infty f(n)[/spoiler] the number which comes closer and closer (with respect to the distance function on R) to the partial sums <span class="math">\sum_{n=1}^m f(n)[/spoiler], as m goes towards infinity"
then this uses the limit and is hence also part of an analytic theory.
You're right in that it gives you more power to separate different expression from another.

-1/12 is the second Bernoulli number B2=1/6 times -1/2! and those often show up when you relate discrete quantities with their smooth interpolations.
For example, the linearization of the exponential function at 0 is "exp(0)·h" and the finite difference is "exp(0+h)-exp(0)". Now sure enough, you get the Taylor expansion

<span class="math">h/(exp(h)-1) = 1 - h/2 + h^2/12 + ...[/spoiler]

More dramatically, there is this equation relating sums and integrals due to Gauss:

<span class="math">\int_a^b f(n) dn = \sum_{n=a}^{b-1} f(n)+(lim_{x\to b}-lim_{x\to a}) (1/2\,f(x)\,-\,1/12\,f'(x)+...)[/spoiler]

For example, for a=2, b=4 and f(n)=n^2, you have

<span class="math">\int_2^4 n^2=(2^2+3^2)+(1/2)(4^2-2^2)-\frac{1}{12}2(4^1-2^1)[/spoiler]

Now naively

<span class="math">\int_0^\infty f(n) dn = \sum_{n=0}^\infty f(n)-(1/2\,f(0)\,-\,1/12\,f'(0)+...)+lim_{x\to b}(1/2\,f(x)\,-\,1/12\,f'(x)+...)[/spoiler]

In the standard theory, f(n)=n gives
"(undefined int) = (undefined sum) - (-1/12) + (undefined lim)".
Note that we speak of undefined expressions only in the meta-theory, limits are not functions.
The point is now that you get a consistent theory of infinite sums, in your sense, by dropping all other undefined terms. E.g. here
"(the sum over all n) = -1/12".
The theory isn't more or less inconsistent than the one that assigns 2 to
1+1/2+1/4+1/8+...,
but they are not compatible with one another.

>>7097075
You keep bringing that up, but of course the equal signs in the logical theory of natural numbers, the theory of groups and the theories of sets is also not the same.