Wave Propagation in Linear Media | re-examined

6.1 Partitioning the integration interval
6.1 Partitioning the integration interval
Once we know that an in nite integral exists, we can theoretically choose any arbitrary
partition of the interval to calculate the sequence of partial sums. In practice, however, we
are better o to select the subdivision points such that the resulting sequence is amenable
to convergence acceleration. A careful choice is already half the guarantee for a trustworthy
result. Let an example illustrate this pitfall.
We consider the well-known integral
Z 1
0
sin x
x dx = : (6.1)
2
We subdivide the interval at the zeros of the sine and calculate the rst 100 elements of
the sequence of partial sums. To this end, we rst evaluate the contributions of the integrals
Ii = R xi+1
f(x) dx between the partition points and then accumulate them to yield the partial
xi
sums Si = Pi k=0 Ii. We nally plot this alternating sequence.
In[1]:= sequ = Table[NIntegrate[Sin[x]/x,fx,i Pi,(i+1) Pig],
fi,0,100g];
partial = FoldList[Plus,0,sequ];
1.59
1.58
1.57
1.56
1.55
1.54
z1 = ListPlot[partial,PlotStyle->fPointSize[0.012]g];
20 40 60 80 100
In[2]:= SequenceLimit[partial] - N[Pi/2]
Out[2]= -15
-1.77636 10
133