By Denis S. Arnon, Bruno Buchberger
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Additional info for Algorithms in Real Algebraic Geometry
Ramanujan himself summed other arctangent series in [lO], [15, p. 421. Glasser and Klamkin [ 1] have summed several arctangent series in an elementary fashion. Further examples of arctangent series are found in Bromwich’s book [l, pp. 314-3153. Example 3. Example 4. ,(l-$)=&cosh($) Examples 3 and 4 constitute the second problem that Ramanujan submitted to the Journal ofthe Zndian Muthematical Society , [ 15, p. 3221. In a later paper , [15, pp. 50-521, Ramanujan studied the more general product In Entry 12, Ramanujan presents a method for approximating the root z0 of smallest modulus of the equation 5 A,zk = 1.
Method. ” If m and n are to be taken as real, then Ramanujan’s remark is pointless, for then this ratio may be made to take any real value. On the other hand, if m and n are to be understood as positive, then Ramanujan’s assertion is false. Ramanujan’s claim would be valid if the hmit L were always between two consecutive convergents. However, this may not be true. For example, the last three convergents 13/33, 33184, and 84/214 given by Ramanujan in Example 4 satisfy the inequalities 13133 > 33184 > 84/214.
This is probably the method that Ramanujan employed to estimate H. 48547086.. . In any event, the estimate of 7$ for H is not as good as Ramanujan would lead us to believe. Entry 7. Let n > 0 and suppose that r is a natural number. Then r-l k=O 2 1)’ )=A(n2+L+l)’ = A( (n+2k+ Proof. The proof is very briefly sketched by Ramanujan. 1) 36 2. 2) completes the proof. Corollary. 3) Proof. 2). Example 1. 4) where p(n) = TC if n < ($ - 1)/2 and p(n) = 0 otherwise. Proof. The proof is sketched by Ramanujan.