Geometric sums

In our work in Section, we learned how to find a degree \(n\) polynomial approximation centered at a value \(a\) for a given function \(f\) with at least \(n\) derivatives. By working with several different functions and \(n\)-values, we've seen that increasing the degree of the polynomial improves the approximation, and also often helps us to see a pattern in the coefficients of the Taylor polynomials.

For example, the degree \(5\) Taylor approximation of \(f(x) = \ln(1+x)\) at \(a = 0\) is \[\begin{aligned}\end{aligned}\] and we see a pattern in the coefficients that allows us to easily generate \(T_6(x)\), \(T_{10}(x)\), or indeed \(T_n(x)\) for any \(n\). Note that if we want to use \(T_{10}(x)\) to estimate \(\ln(\frac{3}{2}) = \ln(1 + \frac{1}{2})\), we need to compute the sum of \(10\) terms given by \[\begin{aligned}\end{aligned}\] This computation suggests at least two questions: is there an easy way (without a computer) to determine the exact value of the \(10\)-term sum \[\begin{aligned}\end{aligned}\], and is there a way we can make sense of continuing the sum indefinitely, \[\begin{aligned}\end{aligned}\]

In this section, we investigate a special collection of similar sums that are called geometric.

Sums such as \[\begin{aligned}\end{aligned}\], \[\begin{aligned}\end{aligned}\], and \[\begin{aligned}\end{aligned}\] all share a similar structure: each has exactly \(n\) terms, and the next term in each sum is found by multiplying the last term by the same number. In the first sum, each subsequent term is found by multiplying by \(\frac{1}{2}\); in the second sum, by multiplying by \(\frac{1}{4}\); in the third sum, the multiplier is \(\frac{-2}{3}\). These sums each have the form \[\begin{aligned}\end{aligned}\], which we call a finite geometric series with ratio \(r\). It turns out that the value of each sum that has this form can be computed quickly without having to add all of the individual terms.

The ideas in Activity can be extended to the general case for any value of \(a\) and any value of \(r \ne 1\). In particular, replacing \(1\) with \(a\) and \(\frac{2}{5}\) with \(r\), our work shows that by finding \(r \cdot S_n\) and then subtracting that quantity from \(S_n\), we get \[\begin{aligned}\end{aligned}\], so that \[\begin{aligned}\end{aligned}\], and therefore \[\begin{aligned}\end{aligned}\]. We summarize this result formally as follows.

A finite geometric series \(S_n\) is a sum of the form \[\begin{aligned}\end{aligned}\], where \(a\) and \(r\) are real numbers such that \(r \ne 1\). The exact value of the finite geometric series \(S_n\) can be computed directly as \[\begin{aligned}\end{aligned}\].

Condensed — the full section is in Boelkins, Active Calculus.

Our initial introduction to sums with many terms came from Taylor polynomial approximations such as \[\begin{aligned}\end{aligned}\]. Because the approximation gets better as we add more terms, it's natural to think about the possibility of the sum extending forever. We begin by asking this question for a finite geometric series such as \[\begin{aligned}\end{aligned}\]: what happens if we let the sum continue indefinitely?

Example demonstrates the general principle that we use to determine if any infinite series has a finite value: we consider the partial sum, \(S_n\), which is the finite sum of the first \(n\) terms, and then investigate whether the partial sums converge to a single value as \(n\) increases without bound. For geometric series, determining whether the partial sums converge or not is straightforward.

Condensed — the full section is in Boelkins, Active Calculus.

If we take \(a = 1\) in Equation and Equation, we see that \[\begin{aligned}\end{aligned}\], provided that \(|r| \lt 1\). To study this equation further, we are going to let \(r\) vary and thus we introduce the function \(f(x) = \frac{1}{1-x}\) and replace \(r\) by \(x\) in Equation to have \[\begin{aligned}\end{aligned}\], which is valid for \(|x| \lt 1\). One reason this equation is interesting is that we have a function \(f(x)\) that can be represented in two different ways: as the rational function \(\frac{1}{1-x}\), and as the infinite polynomial function \(1 + x + x^2 + \cdots\).

For \(x\) such that \(|x| \lt 1\), we know from Equation that the infinite geometric series \(1 + x+ x^2 + \cdots + x^{n-1} + x^n + \cdots\) converges. Thus, it follows that partial sums of the series will approximate its value, which means \[\begin{aligned}\end{aligned}\]. Since the series converges for \(|x| \lt 1\), this also means the approximate equality in holds for \(x\) near \(a = 0\). This result reminds us of approximations generated by Taylor polynomials of degree \(n\).

Moreover, because the infinite series converges, the larger the value of \(n\), the better the approximation will be. Plotting the function \(f(x) = \frac{1}{1-x}\) along with several of the polynomials that arise for different choices of \(n\), say \(T_1(x) = 1 + x\), \(T_4(x) = 1 + x + \cdots + x^4\), and \(T_7(x) = 1 + x + \cdots + x^7\), we see the impact of increasing the degree \(n\) in Figure.

In particular, we observe that as the degree of the polynomial approximation increases, the polynomial not only appears closer to \(f(x) = \frac{1}{1-x}\), but does so on a wider interval of \(x\)-values. Since the infinite series only converges when \(|x| \lt 1\) and the function \(f(x) = \frac{1}{1-x}\) is undefined when \(x = 1\), we also expect the approximations to only be accurate on an interval that lies within \(-1 \lt x \lt 1\). This also reminds of our earlier work with Taylor polynomials where in images such as Figure, increasing the degree of the Taylor polynomial similarly improves the approximation.

In our recent preceding work, we arrived at the function \(f(x) = \frac{1}{1-x}\) by starting with the infinite geometric series \[\begin{aligned}\end{aligned}\] and exploring its partial sums. Next, we change perspective and start with function \(f(x) = \frac{1}{1-x}\) and determine the Taylor polynomial approximations to \(f\) that are centered at \(a = 0\) in order to see an interesting connection.

• A finite geometric series \(S_n\) is a sum of the form \[\begin{aligned}\end{aligned}\], where \(a\) and \(r\) are real numbers such that \(r \ne 1\). The value of the finite geometric series \(S_n\) can be computed directly as \[\begin{aligned}\end{aligned}\].

• An infinite geometric series \(S\) is a sum of the form \[\begin{aligned}\end{aligned}\], where \(a\) and \(r\) are real numbers. If \(|r| \lt 1\), then the infinite geometric series converges to the finite value \[\begin{aligned}\end{aligned}\]. If \(|r| \ge 1\), then the infinite geometric series does not converge to a finite sum.

• If we consider the infinite geometric series with \(a = 1\) and \(|r| \lt 1\) and replace \(r\) with \(x\), we can say that \[\begin{aligned}\end{aligned}\]. As we found in Activity, the Taylor series centered at \(a = 0\) for \(f(x) = \frac{1}{1-x}\) is precisely the infinite geometric series \[\begin{aligned}\end{aligned}\] and we know this series is equal to \(f(x)\) for \(|x| \lt 1\).