
Certain infinite series can be studied using improper integrals.

In order to study the convergence of a series $\sum _{k=k_0}^{\infty } a_k$, our first attempt to determine whether the series converges is to form the sequence of partial sums $\{s_n\}_{n=k_0}$ since we know that the series $\sum _{k=k_0}^{\infty } a_k$ converges if and only if $\lim _{n \to \infty } s_n$ exists. In the case of geometric or telescoping series, we were able to find an explicit formula for $s_n$, and analyze $\lim _{n \to \infty } s_n$ by explicit computation. However, we cannot always find such an explicit formula, and when this is the case, we try to use properties of the terms in the sequence $\{a_n\}$ to determine whether $\lim _{n \to \infty } s_n$ exists. Our first result was the divergence test, which states

If $\lim _{n \to \infty } a_n \neq 0$, then $\sum _{k=k_0} a_k$ diverges.

However, there are still some divergent series that the divergence test does not pick out! We begin this section with such an example that shows how there is a connection between certain special types of series and improper integrals.

Now, let’s take a step back and see what we really needed in the previous example.

• We needed to find a function for which the area under the curve over any particular interval $[n,n+1]$ was less than the area of the rectangle whose height is $a_k$ to establish a lower bound for each $s_n$. Note that we can always do this if $f(x)$ is eventually positive and decreasingincreasing since we may view each $a_k$ as the area of the rectangle that coincides with $f(x)$ at its lefthandrighthand endpoint.
• We needed the function to be “eventually continuous” so the improper integral $\int _{a}^{\infty } f(x) \d x$ can be computed as the limit of a single definite integral.

By “eventually” above, we really mean that $f(x)$ should be continuous, positive, and decreasing on some interval $[a,\infty )$ for some $a>0$; it doesn’t need to happen right away, but it should hold for all real large enough $x$-values. This leads us to an interesting observation.

Let $f(x)$ be an eventually continuous, positive, and decreasing function with $a_k = f(k)$. If $\int _1^\infty f(x) \d x$ diverges, so does $\sum _{k=1}^\infty a_k$.

That’s a pretty good observation, but we can do even better.

Now, let’s take a step back and see what we really needed in the this example.

• We needed to find a function for which the area under the curve over any particular interval $[n,n+1]$ was greater than the area of the rectangle whose height is $a_k$ to establish a lower bound for each $s_n$. Note that we can always do this if $f(x)$ is eventually positive and decreasingincreasing since we may view each $a_k$ as the area of the rectangle that coincides with $f(x)$ at its lefthandrighthand endpoint.
• We needed to establish that the sequence of partial sums is eventually increasing. This must happen if all of the $a_k$ are positivenegative .
• We needed the function to be “eventually continuous” so the improper integral $\int _{a}^{\infty } f(x) \d x$ can be computed as the limit of a single definite integral.

By “eventually” above, we really mean that $f(x)$ should be continuous, positive, and decreasing on some interval $[a,\infty )$; it doesn’t need to happen right away, but it should hold for all real large enough $x$-values. This leads us to an interesting observation.

Let $f(x)$ be an eventually continuous, positive, and decreasing function with $a_k = f(k)$. If $\int _1^\infty f(x) \d x$ converges, so does $\sum _{k=1}^\infty a_k$.

The Integral Test

The observations from the previous examples give us a new convergence test called the integral test:

The next examples synthesizes some concepts we have seen thus far.

p-Series

A very important type of series for future sections is $\sum _{k=1}^\infty \frac {1}{k^p}$, where $p>0$. We call a series that can be brought into this form a $p$-series. We want to determine for which values of $p$ these series converge and diverge.

Notice that in our model examples, both series were $p$-series.

• The harmonic series $\sum _{k=1}^{\infty } \frac {1}{k}$ is a $p$-series with $p = \answer {1}$. It convergesdiverges .
• The series $\sum _{k=1}^{\infty } \frac {1}{k^2}$ is a $p$-series with $p = \answer {2}$. It convergesdiverges .

This result is important enough to list as a theorem.

Which of the following series converge? (Select all that apply)
$\sum _{k=2}^{\infty } \frac {1}{\sqrt {k}}$ $\sum _{k=3}^{\infty } \frac {1}{k^{3/2}}$ $\sum _{k=5}^{\infty } k^{-4}$ $\sum _{k=1}^{\infty } \frac {1}{k^{.1}}$