Sum of an Infinite Geometric Series

Can an infinite sum be finite?

Yes! If each term is a fixed fraction of the previous one, and that fraction $|r|$ is less than 1, the terms shrink so fast that their infinite sum converges to a finite number.

The formula

For an infinite geometric series with first term $a_1$ and common ratio $|r| < 1$:

$S_\infty = \frac{a_1}{1 - r}$

If $|r| \geq 1$, the series diverges (sum is infinite).

Step-by-step: Find $1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots$

Step 1 — Identify $a_1$ and $r$.

$a_1 = 1$. Common ratio: $r = \frac{1/3}{1} = \frac{1}{3}$.

Step 2 — Check convergence. $|r| = 1/3 < 1$ ✓ (the series converges).

Step 3 — Apply the formula:

$S = \frac{1}{1 - 1/3} = \frac{1}{2/3} = \frac{3}{2} = 1.5$

Step 4 — Verify with partial sums:

$S_1 = 1$, $S_2 = 1.333$, $S_3 = 1.444$, $S_4 = 1.481$, $S_5 = 1.494$, ... approaching $1.5$ ✓.

Why it works intuitively

Each term adds less than the previous: $1, 0.333, 0.111, 0.037, \ldots$ The amounts shrink so fast that even adding infinitely many of them can't exceed $3/2$.

Common exam variations

• Repeating decimals: $0.333\ldots = 3/10 + 3/100 + 3/1000 + \cdots = \frac{3/10}{1 - 1/10} = \frac{1}{3}$.
• Bouncing ball: A ball dropped from height $h$ bounces to $rh$ each time. Total distance = $h + 2h \cdot \frac{r}{1-r}$.

Try it in the visualization

Adjust $a_1$ and $r$. Watch the shrinking bars accumulate. The convergence graph shows partial sums approaching the limit. The limit line confirms the formula's prediction.