Topic 10.2 Notes – Working with Geometric Series

1. What a Geometric Series Is

A geometric series is an infinite series where each term is obtained by multiplying the previous term by the same number.

Two common forms you’ll see:

$$\sum_{n=0}^{\infty} ar^n \quad \text{and} \quad \sum_{n=1}^{\infty} ar^{\,n-1}$$

• $a$ = first term
• $r$ = common ratio
• Each term = previous term × $r$

Example pattern:

$$5 + 5\left(\tfrac{1}{4}\right) + 5\left(\tfrac{1}{4}\right)^2 + \cdots$$

Here:

• $a = 5$
• $r = \tfrac{1}{4}$

A quick reminder: a series is the sum of infinitely many terms. We’re asking whether that sum approaches a finite number (converges) or blows up (diverges).

Fast Recognition Trick

If you compute

$$\frac{\text{term}_{n+1}}{\text{term}_n}$$

and always get the same constant, it’s geometric.

If the ratio changes, it’s not geometric. Factorials, powers of $n$, alternating complicated expressions - those are not geometric.

2. The Geometric Series Convergence Rule

Here’s the rule that drives everything:

• If $|r| < 1$, the series converges
• If $|r| \ge 1$, the series diverges

That’s the entire test.

Why the absolute value?

Because what matters is whether the terms shrink toward zero.

• If $r = \tfrac{1}{2}$, terms get smaller.
• If $r = -\tfrac{1}{2}$, they alternate signs but still shrink.
• If $r = 2$, terms grow.
• If $r = -3$, they alternate but grow in size.

The size of $r$ controls everything.

If $|r| < 1$, the series converges.
If $|r| > 1$, the series diverges.
If $r = \pm 1$, the series also diverges.

If $|r|$ is inside that open interval, you’re good. Otherwise, no sum.

On tests, many questions are just: identify $r$, check $|r|$, state converges or diverges.

3. The Sum Formula (When It Converges)

If $|r| < 1$, then:

$$\sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r}$$

Same formula works for $\sum_{n=1}^{\infty} ar^{n-1}$. The index shift does not change $a$.

Quick Example

$$\sum_{n=0}^{\infty} 8\left(\tfrac{3}{5}\right)^n$$

• $a = 8$
• $r = \tfrac{3}{5}$
• $|r| < 1$, so it converges.

Sum:

$$\frac{8}{1 - \frac{3}{5}} = \frac{8}{\frac{2}{5}} = 20$$

That’s the exact value of the infinite sum.

One common mistake: plugging into the formula before checking $|r| < 1$. If it diverges, the formula does not apply.

4. Rewriting a Series to See If It’s Geometric

Often the series is disguised.

Given as a list

Example:

$$12 - 6 + 3 - \tfrac{3}{2} + \cdots$$

Check ratios:

$$\frac{-6}{12} = -\tfrac{1}{2}, \quad \frac{3}{-6} = -\tfrac{1}{2}$$

Constant ratio. So:

• $a = 12$
• $r = -\tfrac{1}{2}$

Since $|r| < 1$, it converges.

Given in sigma form but messy

Example:

$$\sum_{n=1}^{\infty} 4\left(-\frac{2}{3}\right)^{n-1}$$

It already matches $ar^{n-1}$.

• $a = 4$
• $r = -\tfrac{2}{3}$

Sometimes you’ll need to:

• Factor constants out
• Rewrite something like $9 \cdot 3^{-n}$ as $9\left(\tfrac{1}{3}\right)^n$

If you can rewrite it as:

$$\text{constant} \cdot (\text{constant})^n$$

you’re in geometric territory.

5. Common Mistakes and Fast Checks

• Forgetting absolute value
  If $r = -2$, then $|r| = 2$. Diverges.
• Misidentifying $a$
  If the series starts at $n=1$, plug in $n=1$ to find the first term if you’re unsure.
• Using the formula on a divergent series
  If $|r| \ge 1$, stop immediately.
• Assuming every exponential-looking series is geometric
  If the exponent involves $n^2$, factorials, or changing bases, it’s not geometric.

On multiple choice, geometric series are often quick points. On FRQs, they show up when building power series later. You’re expected to recognize them instantly.