Geometric Series

The central idea

A geometric series has the form $\sum_{n=0}^{\infty}ar^n$ or an index-shifted equivalent. It converges exactly when $|r|<1$, in which case its sum is $a/(1-r)$, where $a$ is the first included term. If $|r|\ge1$ and $a\ne0$, it diverges.

Definitions, hypotheses, and notation

For $r\ne1$, the finite identity $S_N=a(1-r^N)/(1-r)$ follows by subtracting $rS_N$ from $S_N$; when $r=1$, $S_N=Na$. The infinite formula is its limit, so it is valid only when $r^N\to0$. This derivation explains both the convergence condition and why substituting an inadmissible ratio into $a/(1-r)$ creates a meaningless answer.

Recurring decimals are geometric series: $0.272727\ldots=0.27+0.0027+\cdots$ has ratio $0.01$. Applications in finance and decay use the same structure, but units and starting time determine which payment or amount is the first term.

Conceptual meaning

Each term is a fixed multiple of the previous one. When $|r|<1$, the unadded tail shrinks geometrically. Negative $r$ alternates signs; the convergence condition depends on magnitude, while the sum formula retains the sign.

A dependable method and decision rule

1. Identify the first term actually included and common ratio.
2. Confirm the ratio is constant by dividing consecutive terms.
3. For a finite sum, use $S_N=a(1-r^N)/(1-r)$ when $r\ne1$, or $S_N=Na$ when $r=1$.
4. Check $|r|<1$ before using $S=a/(1-r)$ for an infinite sum.
5. For a tail beginning later, recompute its first term rather than reusing the original $a$.

Fully worked example

Common mistakes and why they fail