Notes on geometric series

Definition

A geometric series is a sum of the form

\( a + ar + ar^2 + … + ar^k \) in the finite case

or

\( a + ar + ar^2 + … \) in the infinite case.

In both cases, \(a\) and \(r\) are constants. Some people just use the term geometric series to refer to the infinite case. Always be alert for context.

The sum of a finite geometric series with last term of power k is \(a\frac{1-r^{k+1}}{1-r}\)

Step by step:

Let \(S_k = a + ar + ar^2 + … + ar^k \) for some finite k.

If you have been away from mathematics for a while, recall that by basic rules of exponentiation, \(r^0 = 1\) and \(r^1 = r\), so that the above can be rewritten as:

\[ S_k = ar^0 + ar^1 + ar^2 + … + ar^k \tag{1} \]

When you are learning or searching for a useful pattern, don’t hesitate to make all of the pieces explicit. Eventually you will not need to do so, but there is little to lose from extra clarity when you are starting out.

Multiply both sides of (1) by r:

\[ \begin{align} rS_k &= r(ar^0) + r(ar^1) + r(ar^2) + … + r(ar^k) \\ &= (ar^0)r + (ar^1)r + (ar^2)r + … + (ar^k)r \\&= (ar^0)r^1 + (ar^1)r^1 + (ar^2)r^1 + … + (ar^k)r^1 \\ &= ar^{0+1} + ar^{1+1} + ar^{2+1} + … + ar^{k+1} \\ &= ar^1 + ar^2 + ar^3 + … + ar^{k+1} \tag{2} \end{align}\]

Compare (1) and (2). Notice that \(S_k\) and \(rS_k\) share all of their terms except for the first term of \(S_k\) (namely, \(ar^0\)), and the last term of \(rS_k\) (namely, \(ar^{k+1}\)). So if you subtract \(rS_k\) from \(S_k\), all of the middle terms will cancel:

\[ \begin{align} S_k – rS_k = ar^0 &+ ar^1 + ar^2 + … + ar^k \\ &-ar^1 \,- ar^2 – … \,- ar^{k} – ar^{k+1} \end{align} \]

This will leave you with:

\[ S_k – rS_k = ar^0 – ar^{k+1} \]

Factor \(S_k\) out of the left side:

\[ S_k(1-r) = ar^0-ar^{k+1} \]

Factor a out of the right side:

\[ S_k(1-r) = a(r^0-r^{k+1}) \]

Divide both sides by \(1-r\):

\[ S_k = a\frac{r^0-r^{k+1}}{1-r} \]

And now can just rewrite the \(r^0\) as 1 for simplicity:

\[ S_k = a\frac{1-r^{k+1}}{1-r} \]

And there you have it.

Note that k in this formula is the value of the last exponent in the initial series. The first term in that series has an exponent of 0, so the total number of terms is one more than the value of the last exponent. I have found it easier to keep track of this by understanding the steps above than by attempts at rote memorization of formulas, especially since different treatments will set things up slightly differently and end up with formulas that appear slightly different.

What if there is no constant?

Suppose we have a sum of the form \( r + r^2 + … + r^n \). This looks sort of like a finite geometric series, but there is no \(a\)—no constant—so what do we do?

Actually, there is a constant. It’s just hidden away a bit. The constant is r.

Notice: \( r + r^2 + …. + r^n = r + r(r^1) + r(r^2) + …. + r(r^{n-1}) \). The latter has the form of a finite geometric series with \(a = r \) and \( k = n-1 \). So we can plug those into the summation formula and get

\[ \begin{align} S_{n-1} &= r\frac{1-r^{(n-1)+1}}{1-r} &= r\frac{1-r^n}{1-r} \end{align} \]