procedure euler (fct,sum,eps,tim); value eps,tim; integer tim; real procedure fct; real sum,eps; comment euler computes the sum of fct(i) for i from zero up to infinity by means of a suitably refined euler transformation. The summation is stopped as soon as tim times in succession the absolute value of the terms of the transformed series are found to be less than eps. Hence, one should provide a function fct with one integer argument, an upper bound eps, and an integer tim. The output is the sum sum. euler is particularly efficient in the case of a slowly convergent or divergent alternating series; begin integer i,k,n,t; array m[0:15]; real mn,mp,ds; i ≔ ; t ≔ ; m[0] ≔ ct(0); sum ≔ [0]/2; nextterm: i ≔ +1; mn ≔ ct(i); for k ≔ step 1 until n do begin mp ≔ mn+m[k])/2; m[k] ≔ n; mn ≔ p end means; if (abs(mn)<abs(m[n])) ∧ (n<15) then begin ds ≔ n/2; n ≔ +1; m[n] ≔ n end accept else ds ≔ n; sum ≔ um+ds; if abs(ds) < eps then t ≔ +1 else t ≔ ; if t < tim then goto nextterm end euler;