Re: A Formula for Pi

Jose Carlos Santos wrote:

On 20-06-2008 7:16, Maury Barbato wrote:

I found in the book "The Penguin Dictionary of
Curious and Interesting Numbers" by Wells the
following formula involving pi

(pi - 3)/4 =

= sum_{k=1 to infty}[(-1)^(k+1)]/[2k*(2k+1)*(2k+2)]

Is there anybody who knows a proof of this
wonderful
series?

You already got a reply. I only want to remark that
the formula is
equivalent to

pi - 3 = sum_{k = 1 to oo}(-1)^{k + 1}/(k(2k +
2k + 1)(k + 1)

= sum_{k = 1 to oo}(-1)^{k + 1}/(1^2 +
+ 1}/(1^2 + 2^2 + ... + k^2).

Best regards,

Jose Carlos Santos

A little slip. We have

pi - 3 =
sum_{k = 1 to oo}(-1)^{k + 1}/[k(2k+ 1)(k + 1)]=
= (1/6)* sum_{k = 1 to oo}(-1)^{k + 1}/(1^2 + 2^2 + ... + k^2)

It's a very very beautiful formula!
Thank you for your attention.
My Best Regards,
Maury Barbato
.

Relevant Pages

  • Re: A Formula for Pi
    ... Curious and Interesting Numbers" by Wells the ... Best regards, ... Jose Carlos Santos ... How many terms do I have to sum to get 100 decimal ...
    (sci.math)
  • Re: An incompetent plagiarist
    ... Best regards, ... Jose Carlos Santos ... schools and department chairs and president of the school. ... the former President of the college where I teach ...
    (sci.math)
  • Re: measure
    ... > Bobbine wrote: ... > Best regards, ... > Jose Carlos Santos ... the intersection of a decreasing sequence of open ...
    (sci.math)
  • Re: A question regarding Integrable functions
    ... Best regards, ... Jose Carlos Santos ... how about a bit of relaxation to the above (falsy) Theorem ...
    (sci.math)
  • Re: Are functions with null derivative locally constant?
    ... to be affirmative if U is assumed to be an open set. ... Best regards, ... Jose Carlos Santos ... No, in fact there is a strictly increasing function on R, differentiable ...
    (sci.math)