Geometric Series: Convergence to a Limit

Infinite geometric series

If the common ratio $|r| < 1$, the terms shrink toward zero and the infinite sum converges to a finite number:

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

Step-by-step: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$

$a_1 = 1$, $r = 1/2$. Since $|r| = 0.5 < 1$, the series converges.

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

The partial sums approach 2: $S_1 = 1$, $S_2 = 1.5$, $S_3 = 1.75$, $S_4 = 1.875$, ...

Key condition: $|r| < 1$

If $|r| \geq 1$, the terms don't shrink and the sum diverges (goes to infinity).

Applications

Repeating decimals: $0.333... = 3/10 + 3/100 + ... = \frac{3/10}{1 - 1/10} = 1/3$.

Try it in the visualization

Adjust $a_1$ and $r$. Shrinking bars accumulate. The convergence graph shows partial sums approaching the limit. For $a = 1, r = 1/2$: $S = 1/(1-0.5) = 2$. Each term is half the previous, and the partial sums approach 2 but never exceed it.