Thread: Finding the convergence of the infinite series

1. 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?

2. 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?
you asked for it ...

Fermat's Last Theorem: The Basel Problem

3. I'll look that over, but is there an easier way to do it?

4. 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...

5. 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

6. 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.

7. 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?