8.2: Introduction to Infinite Series

Partial Sums

If we approach infinite series in a manner similar to improper integrals, then we will need to do a couple of things.

1. 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. We might use n for the series “ending index.”
2. 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.
3. Apply a limit as t (or n in the case of infinite series) goes off to infinity!

Visualizing the Sequence of Partial Sums

Since we’ll think about an infinite series \(\sum_{k=0}^{\infty} a_k\) as the sequence of its partial sums, \(\left\{\sum_{k=0}^n a_k\right\}_{n=0}^{\infty}\), then we can think about visualizing an infinite series as really the same thing as visualizing a sequence in general (Graphing Sequences).

Let’s consider, as a first (visual) example, the infinite series:

\[\sum_{k=0}^{\infty} \dfrac{3}{k+1} \nonumber \]

We can think about the two important sequences that we’ll consider:

\[\underbrace{\left\{\dfrac{3}{n+1}\right\}_{n=0}^{\infty}}_{\left\{a_n\right\} \text {, the sequence of terms }} \quad \text { and } \quad \underbrace{\left\{\sum_{k=0}^n \dfrac{3}{k+1}\right\}_{n=0}^{\infty}}_{\left\{S_n\right\}\text{, the sequence of partial sums }} \nonumber \]

We can plot the sequence of terms, \(\left\{\dfrac{3}{n+1}\right\}_{n=0}^{\infty}\), and visualize the limit \(\lim _{n \rightarrow \infty} a_n=0\). This sequence of terms converges to 0.

Figure 8.2.4 \(\left\{a_n\right\}\), the sequence of terms in the series.

Then, we can compare this with the plot of the partial sums, \(\left\{S_n\right\}\) where:

\[S_n=\sum_{k=0}^n \dfrac{3}{k+1} \nonumber \]

Figure 8.2.5 \(\left\{S_n\right\}\), the sequence of partial sums for the series.

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 8.2.5 is the accumulation of the heights of the preceding points in Figure 8.2.4. We can visualize this to make it easier by overlaying some information onto the plot of partial sums in Figure 8.2.5.

Figure 8.2.6 \(\left\{S_n\right\}\) and \(\left\{a_n\right\}\) visualized together.

Hopefully this does a good job of illustrating the connections between the two:

\[\begin{aligned}
S_0 & =a_0 \\
S_1 & =a_0+a_1 \\
& =S_0+a_1 \\
S_2 & =a_0+a_1+a_2 \\
& =S_1+a_2 \\
S_3 & =a_0+a_1+a_2+a_3 \\
& =S_2+a_3 \\
\vdots & \\
S_n & =a_0+a_1+\ldots+a_n \\
& =S_{n-1}+a_n
\end{aligned} \nonumber \]

In order to further examine the connection between sequences and partial sums, let’s look at an interactive example, where you can create the terms of a sequence (however you’d like) and examine the corresponding partial sums.

Special Series

Let’s look at three examples where we can think about partial sums and play with our new idea of series convergence.

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 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.

Practice Problems