Mma prefers

Pi/8 + ArcCot[2] + ArcCot[3]/2

 
----- Original Message -----
From: "R. William Gosper" <rwg@spnet.com>
To: <math-fun@mailman.xmission.com>
Sent: Monday, February 24, 2003 9:17 PM
Subject: Re: [math-fun] identity from a stranger

> Assuming he(?) means                                                                            
>   inf
>   ====       k      
> 1 \        1         1           2         1
> -  >    (- -)  (----------- + ------- + -------),
> 2 /        4    2 (4 k + 3)   4 k + 2   4 k + 1
>   ====      
>   k = 0
>
> my Macsyma series package gets atan(2)/2 + atan(1/2), as would Maple and
> Mma, presumably.  It isn't hard to show this is atan(phi), but neither is
> it an obvious simplification.
> --rwgOC
>
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