Geometric Series

Definition

A geometric series is a series where each term is a constant multiple of the previous term:

$$\sum _{k=0}^{\infty }{x}^k=1+x+x^2+x^3+\dots$$

Intuition

Multiplying the partial sum by $x$ and subtracting from itself cancels all middle terms, leaving a closed-form expression. For $|x|<1$ the terms shrink to zero and the sum converges; for $|x|\ge 1$ they do not.

Formal Description

Multiplying by $x$ and subtracting gives the finite sum formula:

$$(1-x)\sum _{k=0}^n{x}^k=1-x^{n+1}⟹\sum _{k=0}^n{x}^k=\frac{1-x^{n+1}}{1-x}.$$

Taking $n\to \infty$: when $|x|<1$, $x^{n+1}\to 0$, yielding the infinite geometric series:

$$\sum _{k=0}^{\infty }{x}^k=\frac{1}{1-x},(|x|<1).$$

The series diverges for $|x|\ge 1$.

Key Results

• Converges if and only if $|x|<1$, with sum $\frac{1}{1-x}$.
• Useful identity: $\frac{1}{1+x}={\sum }_{k=0}^{\infty }(-x{)}^k=1-x+x^2-x^3+\dots$, $|x|<1$.

Example. $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots =\frac{1}{1-\frac{1}{2}}=2$.

Applications

The geometric series is the foundation for the Ratio Test and for deriving Power Series and Taylor Series of elementary functions such as $\ln (1+x)$ and $\arctan x$.

Trade-offs

The formula $\frac{1}{1-x}$ requires $|x|<1$; at the boundary ($|x|=1$) the series diverges. For complex $x$, convergence still requires $|x|<1$.

Links

• Sequences and Series
• Ratio Test
• Power Series
• Taylor Series