As mentioned in the previous
section, the total sum of a series is often desired, but it is
not trivial to obtain. The knowledge of whether a series is convergent
or divergent becomes important since there is no point trying to determine
the total sum of a series if it is a divergent series. The integral
test is one of the methods which can be used to determine the convergence
of a series, and it is stated as follows:
Let f be a positive, continuous, and decreasing function on [1, )
and a_{n} = f(n), then
(1) The series is
convergent if is
convergent.
(2) The series is
divergent if is
divergent. Note that the improper integral is involved in the integral
test.
Furthermore, if one uses a partial sum (S_{n}) to estimate the
total sum (S) of a series using the integral test, the remainder R_{n} =
S  S_{n} is given by
The above inequalities can be further refined by adding S_{n} to both
sides to yield
where S = R_{n} + S_{n.}
These inequalities give a lower bound and upper bound for the estimation
of the total sum of a given series. It is a common practice to take the
midpoint of the interval for the approximation, and it provides a more
accurate estimation.
For example, take the series (also
known as the pseries). Let f(x) = 1/x^{p}, and it is a positive,
continuous and decreasing function. Applying the integral test to this
series yields
Hence, the pseries converges if p > 1.
