With an integral, we have a nice way of evaluating integrals of nicely behaved functions with finite limits of integration (Fundamental Theorem of Calculus (Part 2)).
Then, when we talked about improper integrals, we built a nice way to think about evaluating integrals with unbounded limits of integration (Evaluating Improper Integrals (Infinite Width)). How will we use this to think about infinite series, a sum of the infinitely many terms from an infinite sequence?
If we approach infinite series in a manner similar to improper integrals, then we will need to do a couple of things.
Truncate the infinite series at some finite ending point. This is what we did with the integral, when we replaced the infinity with some real number variable \(t\text{.}\) We might use \(n\) for the series “ending index.”
Find a formula for this truncated/finite version. For the integrals, we could use the Fundamental Theorem of Calculus (Part 2) for this! For series, we’ll need to do something else.
What do you think this means for the infinite series \(\sum_{k=1}^\infty \left(\frac{9}{10^k}\right)\text{?}\) Does the infinite series converge or diverge?
This is hopefully a nice little introduction to how we’ll think about infinite series: we’ll consider, instead, the sequence of sums where we add up more and more terms. This is also a nice first example, because we really just showed that
For an infinite series \(\displaystyle \sum_{k=1}^\infty a_k\text{,}\) we call \(S_n = \displaystyle\sum_{k=1}^{n} a_k\) the \(n\)th Partial Sum of the infinite series.
We say that the infinite series \(\displaystyle\sum_{k=1}^\infty a_k\) converges to the real number \(L\) if the sequence \(\{S_n\}_{n=1}^\infty\) converges to \(L\) (where \(\displaystyle\lim_{n\to\infty} S_n = L\)), where \(S_n = \displaystyle \sum_{k=1}^n a_k\) is the \(n\)th partial sum of the infinite series.
If the sequence of partial sums \(\{S_n\}_{n=1}^\infty\) diverges (the limit \(\displaystyle\lim_{n\to\infty} S_n\) does not exist), then we say that the infinite series \(\displaystyle\sum_{k=1}^\infty a_k\) diverges.
SubsectionVisualizing the Sequence of Partial Sums
Since we’ll think about an infinite series \(\displaystyle\sum_{k=0}^\infty a_k\) as the sequence of its partial sums, \(\left\{\displaystyle\sum_{k=0}^n a_k\right\}_{n=0}^\infty\text{,}\) then we can think about visualizing an infinite series as really the same thing as visualizing a sequence in general (Graphing Sequences).
We can think about the two important sequences that we’ll consider:
\begin{equation*}
\underbrace{\left\{\frac{3}{n+1}\right\}_{n=0}^\infty}_{\{a_n\}\text{, the sequence of terms}} \hspace{1cm} \text{and}\hspace{1cm} \underbrace{\left\{\sum_{k=0}^n\frac{3}{k+1}\right\}_{n=0}^\infty}_{\{S_n\}\text{, the sequence of partial sums}}
\end{equation*}
We can plot the sequence of terms, \(\left\{\dfrac{3}{n+1}\right\}_{n=0}^{\infty}\text{,}\) and visualize the limit \(\displaystyle \lim_{n\to\infty} a_n=0\text{.}\) This sequence of terms converges to 0.
This image is fine, but not very good at showing how the sequence of terms and the sequence of partial sums are related to each other. We should note that each point in Figure 8.2.4 is the accumulation of the heights of the preceding points in Figure 8.2.3. We can visualize this to make it easier by overlaying some information onto the plot of partial sums in Figure 8.2.4.
We had noted earlier (in Section 8.1) that it was hard to find explicit formulas (or recursion relations) for sequences where we had the first few terms.
This remains true when we think about finding formulas for the sequences of partial sums. Notice that it is easy to find the location of the horizontal asymptote in Figure 8.2.3 (by evaluating \(\displaystyle \lim_{n\to\infty} a_n\)), but that we did not attempt to find one to for the partial sums in Figure 8.2.4 or Figure 8.2.5.
If you’d like to try this, then we need to find a formula for \(S_n\text{.}\) Try to find the first several partial sums by adding up terms in the series. Then try to find a formula to predict the next partial sum. This will definitely not be easy!
Ok, actually, this will be an impossible task. There is no closed-form formula for this. We cannot simply find \(\displaystyle \lim_{n\to\infty}S_n\) in the way that we’ve found the limit of the sequence of terms.
For each of the following series, write out a few of the terms of the series. Then write out the corresponding partial sums. Use these to find a formula for \(S_n\text{,}\) the \(n\)th partial sum. Then make a claim about whether or not the series converges and what it converges to.
This one is tricky! It’s hard to notice anything unless we write out the series term formula a bit differently. Use Partial Fractions to rewrite \(\dfrac{2}{k^2-1}\) as \(\dfrac{1}{k-1} - \dfrac{1}{k+1}\text{.}\)
These examples are a bit misleading: we often won’t be able to do this kind of thing! For most infinite series, we will struggle to find an explicit formula for the \(n\)th partial sum. In these examples, though, we took advantage of some specific structure.
In this first example (as well as the example in Activity 8.2.1), we noticed that because of the exponential function defining the terms, we were able to find some nice patterns in the partial sums. We’ll explore this a bit more later in Section 8.6.
Then in these other two examples, we noticed that once we could write each term as really a difference of two fractions that have a really similar structure, we got these “repeat” values from term to term where the opposite signs made things add up to 0. These are called “telescoping series,” and they’re mostly fun examples to think about partial sum formulas. We’ll see some pop up later though, and Partial Fraction Decomposition is a nice trick to keep in mind for these kinds of things.
For each of the following descriptions of \(S_n\text{,}\) explain what conclusions we can make about the infinite series. Be specific, and justify your answers.
