Wave Propagation in Linear Media | re-examined

6Towards a quadrature routine
Remark (Cumulative sums) Using the function FoldList is a very elegant and fast
way to compute cumulative sums of a given list. FoldList[Plus,x,fa,b,cg] returns
fx,x+a,x+a+b,x+a+b+cg [140]. Normally, x will be set to zero. If the leading zero element
of the result might disturb subsequent calculations, it can be omitted with Drop[list,1].
The numerically found limit of this sequence is exact within the machine precision of 16 digits.
Now we inadvertently skip every other zero in the calculation of the series and integrate
over full instead of half periods. Consequently the series no longer alternates, which has a
catastrophic e ect on the determination of the limit.
In[3]:= sequ = Table[NIntegrate[Sin[x]/x,fx,2 i Pi,2 (i+1) Pig],
fi,0,100g];
partial = FoldList[Plus,0,sequ];
1.5675
1.565
1.5625
1.56
1.5575
1.555
1.5525
z2 = ListPlot[partial,PlotStyle->fPointSize[0.012]g];
20 40 60 80 100
In[4]:= SequenceLimit[partial] - N[Pi/2]
Out[4]= -0.00025589
Remark (Improvement of the result) The result may neither be improved by increasing
the precision of the numerical computations nor by including a larger number of
sequence members in the extrapolation.
The example may be considered rather pathological, but we can con rm the di erences between
half- and full-period intervals also for other logarithmically converging integrals like
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