Geometric Series: The Basics

I was just tutoring someone in their Calculus 2 course, and we did a problem on geometric series. When I was a student, I remember memorizing several facts about geometric series without really grokking the algebraic details of why the formulas you come up with are so nice. Since it is fresh in my mind, I figured I’d go through the standard approach.

I’m going to assume the reader is generally familiar with what a geometric series is. In short, you start with a geometric sequence, which has a starting value $a$ , and each term is found by multiplying the previous term by a constant ratio $r$ . You’ll typically see a geometric sequence written as $a,\ ar,\ ar^2,\dots,ar^n,\dots$

A series is just the sum of a sequence. So a geometric series could be written as shown.

a + ar + ar^2 + \cdots + ar^n + \cdots

Partial sums of a series are found by adding the first several terms. For example, the first partial sum of a geometric series is $a$ , and the third partial sum is $a + ar + ar^2$ . Our goal is to find a formula for the $k^\text{th}$ partial sum of a geometric series, and analyze what it means for an infinite geometric series.

Let’s call the $k^\text{th}$ partial sum $S_k$ . Then, by definition we have

S_k = a + ar + ar^2 + \cdots + ar^k.

To make things clearer, we will divide through by $a$ to get

\frac 1a S_k = 1 + r + r^2 + \cdots + r^k.

Now, multiply both sides by $(1-r)$ , and focus on expanding the right side. Multiplying through by 1 gives each of the original terms on the right; multiplying through by $-r$ gives us all the terms on the right with one higher degree, and with a negative sign.

\begin{aligned} (1-r)(1+r+r^2 + \cdots + r^k)=& 1+ r +r^2 + \cdots + r^k\cdots\\ &- r – r^2 -\cdots – r^k – r^{k+1} \end{aligned}

Thus, all terms from $r$ to $r^k$ cancel, leaving us only with $1-r^{k+1}$ . This means our original equation, after multiplying by $(1-r)$ and rearranging to get $S_k$ alone, becomes

S_k =\frac{ a(1-r^{k+1})}{1-r}.

And this is our general, simplified formula for the partial sums of a geometric series.

We define the value of an infinite series by the limit of its partial sums. In particular, our goal is to determine $\lim\limits_{k\to\infty}S_k$ . Doing so requires some casework on the value of $r$ .

First, assume that $r=1$ . Then this formula is never defined due to division by 0 in the denominator. Of course if $a\neq 0$ and $r=1$ , we get the same nonzero term repeating, so the series diverges. Cool.

Next, assume that $r = -1$ . Then $S_k$ will flip between 0 and $a$ . Since it oscillates, the series never converges.

Now we take $|r|>1$ . In this case, the $r^{k+1}$ term grows without bound, causing the limit to diverge as well.

Thus, the only time we get convergence is when $|r|<1$ . In this final case, we see that $\lim\limits_{k\to\infty}r^{k+1} = 0$ , and thus $\lim\limits_{k\to\infty}S_k = \frac{a}{1-r}.$ Therefore, if a geometric series with first term $a$ and common ratio $r$ converges (i.e. if $|r|<1$ ) then its sum is $\boxed{\frac{a}{1-r}}.$ And there you have it!

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