The geometric series is the most important infinite series you’ll ever meet:

$$\frac{1}{1-x} = 1 + x + x^2 + x^3 + x^4 + \cdots \qquad \text{for } |x| < 1$$

Each term is $x$ times the previous one. The sum has a closed form — a finite expression for an infinite sum.

Deriving the formula

Start with the partial sum:

$$S_N = 1 + x + x^2 + \cdots + x^N$$

Multiply both sides by $x$:

$$xS_N = x + x^2 + x^3 + \cdots + x^{N+1}$$

Subtract:

$$S_N - xS_N = 1 - x^{N+1}$$

$$S_N = \frac{1 - x^{N+1}}{1 - x}$$

If $|x| < 1$, then $x^{N+1} \to 0$ as $N \to \infty$, so:

$$S = \frac{1}{1 - x}$$

If $|x| \geq 1$, the terms don’t shrink and the series diverges.

Why this series is everywhere

The geometric series is a universal tool:

Decimal expansions are geometric series in disguise: $$0.333\ldots = \frac{3}{10} + \frac{3}{100} + \frac{3}{1000} + \cdots = \frac{3/10}{1 - 1/10} = \frac{1}{3}$$

Compound interest: Investing $1$ at rate $r$ for $n$ periods gives $\sum_{k=0}^{n} r^k$.

The Euler product: The key step in connecting the zeta function to primes uses the geometric series for each prime $p$:

$$\frac{1}{1-p^{-s}} = 1 + \frac{1}{p^s} + \frac{1}{p^{2s}} + \frac{1}{p^{3s}} + \cdots$$

This is the geometric series with $x = p^{-s}$. Each factor in the Euler product is a geometric series.

Generating functions: In combinatorics and probability, encoding a sequence $(a_0, a_1, a_2, \ldots)$ as a power series $\sum a_n x^n$ turns convolution into multiplication. The geometric series is the generating function for the constant sequence $(1, 1, 1, \ldots)$.

Explore: partial sums

Drag the slider to change $x$ and watch the partial sums converge (or diverge):

Convergence boundary

At $x = 1$: $1 + 1 + 1 + \cdots$ diverges. At $x = -1$: $1 - 1 + 1 - 1 + \cdots$ oscillates (Grandi’s series — no sum in the usual sense). At $x = 1/2$: $1 + 1/2 + 1/4 + \cdots = 2$. At $x = -1/2$: $1 - 1/2 + 1/4 - \cdots = 2/3$.

The series converges inside the unit disk $|x| < 1$ and diverges outside. The boundary $|x| = 1$ is where things get interesting — and where number theory lives.

Variations

Shifted geometric series (starting from $n=1$): $$\frac{x}{1-x} = x + x^2 + x^3 + \cdots$$

Derivative (gives a weighted series): $$\frac{1}{(1-x)^2} = 1 + 2x + 3x^2 + 4x^3 + \cdots$$

With coefficients $a_n$ (a Dirichlet series uses $n^{-s}$ instead of $x^n$ — a different kind of generating function, central to analytic number theory).