| Title: | Mathematics at DEC |
| Moderator: | RUSURE::EDP |
| Created: | Mon Feb 03 1986 |
| Last Modified: | Fri Jun 06 1997 |
| Last Successful Update: | Fri Jun 06 1997 |
| Number of topics: | 2083 |
| Total number of notes: | 14613 |
Any authorities on convergence/divergence out there?
n -p
lim Sigma x
n->oo x=1
is infinite when p=1, but is finite when p=2. At what value of p
(1<p<2) does the sum of this infinite series border between infinity
and "finity"? What's its largest finite value?
| T.R | Title | User | Personal Name | Date | Lines |
|---|---|---|---|---|---|
| 790.1 | Reimann rolls over in his grave | SQM::HALLYB | Profitus Interruptus | Fri Nov 20 1987 11:15 | 11 |
Isn't this just the Zeta function? Converges for all p > 1,
diverges for all p <= 1.
I believe for any real r > 1 there exists a p such that
n -p
r = lim Sigma x
n->oo x=1
John
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| 790.2 | Finally, one I think I can answer :-) | EAGLE1::BEST | R D Best, sys arch, I/O | Fri Nov 20 1987 11:33 | 18 |
I think the Cauchy integral test can give an answer.
{ I think there are some conditions on s(x) which I can't recall at the moment;
they involve the usual stuff about s being finite over some interval etc.}
|
v
if integral( 0, oo, s(x), x ) exists, then
oo
sigma s( n ) converges.
n = 1
x=oo
integral( 0, oo, x^(-p), x ) = [ x^(1-p)/(1-p) ]| for p<>1.
x=0
= 0 - 0 = 0 for 1-p<0 or p>1.
The region of convergence is (at least) the open infinite interval { p : p>1 }.
I'm rusty on this; can anyone else confirm ?
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| 790.3 | Gamma? | COMICS::DEMORGAN | Richard De Morgan, UK CSC/CS | Mon Feb 15 1988 03:58 | 13 |
Speaking of the Riemann zeta function, what is the following formula called (if anything)? I remember deriving it in high school, but as with most maths theorems I discovered, they had already been discovered several hundred years previously: oo -- gamma = 1 - \ zeta(n) - 1 / ----------- -- n n=2 where gamma is Euler's constant. | |||||