Quote:
Originally Posted by tonybarry
Dear Steven,
Eratosthenes may not be presenting hogwash.
{1+2+3+4+... to infinity} = -1/12
http://www.smithsonianmag.com/smart-...949559/?no-ist
Phil Plait does a good rendition of the series according to Ramanujan. He also offers an apology to the conventional mathematicians among us, who take umbrage at the novel use of = .
Would you care to comment ?
Regards,
Tony Barry |
Hello Barry,
Its interesting that Phil Plait ended up back tracking on this.
http://www.slate.com/blogs/bad_astro...ng_result.html
To put this {1+2+3+4+5+......}= -1/12 into perspective, the series is clearly divergent so mathematicians are looking looking for a corresponding convergent geometric series.
Mathematicians start off with an oscillating series
S1={1-1+1-1+1-.......}
And
S2={1-2+3-4+5-6.....}
If you add S2 to itself with terms shifted one place ie -2+1, 3-2, -4+3 etc
you get S2+S2={1-1+1-1+1-1....}= S1
S2=S1/2
If S3={1+2+3+4+5+6...} and you subtract S2 from S3
You get (0+4+0+8+0+...}=4S3
S3=-S2/3
Now S1={1-1+1-1+1-.......}=1/2, hence S2=1/4 and S3=-1/12 the desired result.
You might wonder where S1={1-1+1-1+1-.......}=1/2 comes from.
Its an oscillating series which is neither divergent or convergent.
In order to calculate this value one needs to compare it to a convergent series.
This is where the analytic continuation comes into the picture involving complex numbers which involves a geometrical interpretation.
The oscillating series sits on a boundary of a circle. A geometric series outside the circle is divergent, a geometric series inside the circle is convergent.
When a convergent geometric series approaches the boundary, its summed value is found to be S1=1/2.
Hence S1=1/2 S2=1/4 and S3=-1/12.
This shows we are not evening summing the series, we are summing a convergent series that is mathematically close to it.
Regards
Steven