Improper Integral Extra Credit
Jarod Hart
Math 121
For any student planning on taking math 122, you will see topics on series. Many of these infinite
series are closely related with some of the improper integrals that we look at in class.
Series
We have touched on the idea of series a little bit in this class already. In short, series are infinite sums.
As it is difficult to deal with and impossible to actually sum an infinite number of terms, we revisit the
familiar idea of a limit. That is a series is an expression of the form
∞
∑
n=1
an=lim
N→∞
N
∑
n=1
an
where anare numbers. In fact we have seen these types of sums in defining the definite integral to be a
limit of Reimann sums
Zb
a
f(x)dx =lim
N→∞
N
∑
n=1
f(xn)b−a
n=
∞
∑
n=1
f(xn)b−a
n.
We use similar definitions with series as described for improperintegrals. That is a series ∑∞
n=1anis said
to converge if
lim
N→∞
N
∑
n=1
an
exists and is finite. The series ∑∞
n=1anis said to diverge if
lim
N→∞
N
∑
n=1
an
is ±∞or does not exist. Here we call SN=∑N
n=1ana partial sum as it is part of the entire infinite sum, and
the above limits are called the limit of partial sums.
1
