Finding the convergence of the infinite series

• Dec 6th 2009, 04:01 PM
stones44
Finding the convergence of the infinite series
Consider the series: SUM (when k=1 to k=infinity) of 1/(k^2). I know the series converges to some value S, but how can I find out what S is?
• Dec 6th 2009, 04:14 PM
skeeter
Quote:

Originally Posted by stones44
Consider the series: SUM (when k=1 to k=infinity) of 1/(k^2). I know the series converges to some value S, but how can I find out what S is?

Fermat's Last Theorem: The Basel Problem
• Dec 6th 2009, 04:26 PM
stones44
I'll look that over, but is there an easier way to do it?
• Dec 6th 2009, 07:19 PM
zhupolongjoe
There is no easy way. We use tests just to know if series converge or diverge. The only series we can compute easily would be like geometric series or Maclaurin series or perhaps telescoping series...
• Dec 6th 2009, 07:26 PM
tonio
Quote:

Originally Posted by stones44
I'll look that over, but is there an easier way to do it?

Here you have 14 proofs: http://secamlocal.ex.ac.uk/people/st.../etc/zeta2.pdf

Tonio
• Dec 7th 2009, 08:19 AM
skeeter
Quote:

Originally Posted by stones44
I'll look that over, but is there an easier way to do it?

... if I knew of an "easier" way, I'd have posted it myself.
• Dec 7th 2009, 06:25 PM
stones44
Well if I am told the 100th partial sum is approx: 1.635, is there a way (that is easier) to show that the convergence value is within thousandths of that number?