Theorem 5.3.1.
If a series \(\sum a_n\) converges, then its terms must converge to zero: \(a_n \to 0\text{,}\) but not conversely!
That is, failure to have \(a_n \to 0\) proves divergence.
Section 5.3 The Divergence and Integral Tests
References.
- Calculus, Early Transcendentals by Stewart, Section 11.3.
Introduction.
Often the first question to be answered about a series \(\sum a_n\) is whether it even converges, giving a numerical value.
Then, since we will usually get values only via the approximation of summing a finite nunber of terms and so using a partial sum \(S_N = \sum_{n=1}^N a_n\) as an approximation of the infinite sum, the next important question is to be able to say somethig about how accurate this approximation is.
In this section we first see a test that is fairly simple, but can only give the “negative” result that a series does not converge; then we meet our first method for showing that some series do converge, and also for saying something about the accuracy of partial sums as approximations.
Subsection 5.3.1 The Divergence Test
There is an important point worth repeating from the previous section: for an infinite sum to converge, it is required that the individual terms go to zero, but this alone is not enough, as shown by the Harmonic Series \(\sum_{n=1}^\infty 1/n\text{,}\) for which the terms \(a_n = 1/n \to 0\) as \(n \to \infty\text{,}\) but the sum diverges (“is infinite”).
The name refelcts the fact that this test can only show divergence; its result is always either
“the series diverges” or “I do not know”.
Subsection 5.3.2 The Integral Test and Estimates of Sums
Most often, the value of a series (infinite sum) \(S=\sum a_n\) will be computed accurately but not exactly by computing a partial sum \(S_N\) for a sufficiently large \(N\text{.}\)
This leaves two important and related questions:
- Does the series converge to a number \(S=\sum_{n=1}^\infty a_n\text{?}\)
- If so, how accurate is a given approximation \(S_N = \sum_{n=1}^N a_n\text{?}\) Or if in pursuit of som eneeded degree of accuracy, you might ask how large does \(N\) need to be to meet a given accuracy requirement \(|S-S_N| \leq \epsilon\text{?}\)
These questions might seem familiar; we will see that they are closely related to the questions of whether an improper integral \(I = \int_1^\infty f(x)\, dx = \lim_{N \to \infty}\int_1^N f(x)\, dx\) converges, and of how close those proper “partial” integrals \(I_N =\int_1^N f(x)\, dx\) are to the value of the improper integral \(I\text{.}\)
Question A, and The Integral Test.
Both questions can be answered for a sequence of positive numbers given by a decreasing function, \(a_n=f(n)\text{,}\) so that the sequence itself is also decreasing.
Loosely speaking, the main observation is this:
\begin{equation}
\int_1^\infty f(x) \, dx \leq \sum_{n=1}^\infty a_n \leq a_1 + \int_1^\infty f(x) \, dx\tag{5.3.1}
\end{equation}
so that either
- both the integral and the sum are convergent (finite values for all three terms above), or
- both are divergent (infinite values for all terms above).
This is seen by considering the sum as the are of a sequence or rectangles along the \(x\)-axis of height \(a_n\text{,}\) width 1 on the interval \([n,n+1]\text{,}\) and comparing this to the areas represented by the improper integral.
Theorem 5.3.2. The Integral Test.
If \(a_n = f(n)\) for \(f\) a positive valued and decreasing function defined for \(x \geq 1\text{,}\) then
- If \(\ds\int_1^\infty f(x) \, dx\) converges, so does \(\ds\sum_{n=1}^\infty a_n.\)
- If \(\ds\int_1^\infty f(x) \, dx\) diverges, so does \(\ds\sum_{n=1}^\infty a_n.\)
Checkpoint 5.3.3.
Determine which of the following sums converges, using the above test.
- \(\displaystyle \ds\sum_{n=1}^\infty \frac{1}{n^2}\)
- \(\displaystyle \ds\sum_{n=1}^\infty \frac{1}{n}\)
- \(\displaystyle \ds\sum_{n=1}^\infty \frac{1}{\sqrt{n}}\)
More generally, what about \(\ds\sum_{n=1}^\infty \frac{1}{n^p}\) for various values of the constant \(p\text{?}\)
Such sums are called \(p\)-series, and are one of our two archetypical types of examples, along with Geometric Series.
Question B, and The integral remainder estimate.
If the Integral Test has shown that a sum converges, we can move onto the second question above, of how close a partial sum \(S_N\) is to the exact value \(S\) of the series.
The basic fact is a modification of (5.3.1) above, using integrals and sums starting at \(N\) and \(N+1\) instead of at \(1\text{,}\) along with the observation that the error or remainder in \(S_N\) as an approximation of \(S\) is
\begin{equation*}
R_N = S-S_N = \sum_{n=1}^\infty a_n - \sum_{n=1}^N a_n = \sum_{n=N+1}^\infty a_n.
\end{equation*}
We get
\begin{equation}
\int_{N+1}^\infty f(x) \, dx \leq R_N
\leq \int_N^\infty f(x) \, dx.\tag{5.3.2}
\end{equation}
The second part is probably the most useful, since it gives a upper bound on the size of the error, if we can evaluate the improper integral.
Checkpoint 5.3.4.
For the \(p\)-series \(\ds\sum_{n=1}^\infty \frac{1}{n^2}\)
- How accurate is the approximation \(\ds\sum_{n=1}^{100} \frac{1}{n^2}\text{?}\)
- What about \(\ds\sum_{n=1}^{10^6} \frac{1}{n^2}\text{?}\)
- If we want a result accurate to within \(10^{-10}\text{,}\) how many terms should we sum?
Study Guide.
Study Calculus Volume 2, Section 5.3 2 ; in particular
- The Definition of p-series
- Theorem 8 The Divergence Test
- Theorems 9 The Integral Test and 10 Remainder Estimate from the Integral Test
- Examples 13, 14, 15, 16
- Checkpoints 12, 13, 14, 15
- and one or several exercises from each of the following groups: 138–147, 152–157, 158–161, 169–172, 178 & 179.
openstax.org/books/calculus-volume-2/pages/5-3-the-divergence-and-integral-testsopenstax.org/books/calculus-volume-2/pages/5-3-the-divergence-and-integral-tests
