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You will observe that this partial sum converges to 1.
NOTE The above summation was performed backwards because this
scheme will ensure a more accurate result and will keep all the significant
digits of the smallest term of the sum.

In-Class Exercises

Compute the following infinite sums:
∞
∑ 1
Pb. 4.16 2 ( k − 2k− 1
k= 1 1 2)
∞
∑ sin(2k − ) 1
Pb. 4.17 (2k −
k= 1 ) 1
∞
∑ cos( )k
Pb. 4.18 4
k 1= k

∞
∑ sin( / )k 2
Pb. 4.19 3
k 1= k
∞
∑ 1
Pb. 4.20 k sin( k)
k= 1 2

4.4 Numerical Integration
The algorithm for integration discussed in this section is the second simplest
available (the trapezoid rule being the simplest, beyond the trivial, is given
at the end of this section as a problem). It has been generalized to become
more accurate and efficient through other approximations, including Simp-
son’s rule, the Newton-Cotes rule, the Gaussian-Laguerre rule, etc. Simp-
son’s rule is derived in Section 4.6, while other advanced techniques are left
to more advanced numerical methods courses.
Here, we perform numerical integration through the means of a Rieman
sum: we subdivide the interval of integration into many subintervals. Then
we take the area of each strip to be the value of the function at the midpoint
of the subinterval multiplied by the length of the subinterval, and we add the

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