A geometric series is the sum of the terms of a geometric sequence. Because each term is obtained by multiplying by a constant common ratio \(r\), the series has a predictable structure that leads to closed-form sum formulas.
Definition If \(\{a_n\}\) is a geometric sequence with first term \(a_1\) and common ratio \(r\), then the corresponding geometric series is: \[ a_1+a_1r+a_1r^2+\cdots. \] The finite geometric series (first \(n\) terms) is \[ S_n=a_1+a_1r+a_1r^2+\cdots+a_1r^{n-1}. \]
Finite geometric series: the partial sum \(S_n\)
The key idea is to multiply the sum by \(r\) and subtract to cancel most terms.
Derivation (standard algebra method)
\[ S_n=a_1+a_1r+a_1r^2+\cdots+a_1r^{n-1}. \]
\[ rS_n=a_1r+a_1r^2+a_1r^3+\cdots+a_1r^n. \]
Subtract the second equation from the first. All middle terms cancel:
\[ S_n-rS_n=a_1-a_1r^n. \]
\[ S_n(1-r)=a_1(1-r^n). \]
\[ S_n=a_1\cdot\frac{1-r^n}{1-r}\quad (r\ne 1). \]
Special case If \(r=1\), every term equals \(a_1\) and the sum is: \[ S_n=n\cdot a_1. \]
Worked example: \(3+6+12+24+48\)
This is a geometric series with \(a_1=3\), \(r=2\), and \(n=5\).
\[ S_5=3\cdot\frac{1-2^5}{1-2} =3\cdot\frac{1-32}{-1} =3\cdot 31 =93. \]
Infinite geometric series \(S_\infty\) and convergence
An infinite geometric series adds infinitely many terms: \[ S_\infty=a_1+a_1r+a_1r^2+\cdots \] Such a series has a finite sum only if the terms shrink toward \(0\), which happens when \(|r|<1\).
Infinite sum If \(|r|<1\), then \(r^n\to 0\) as \(n\to\infty\), and the infinite geometric series converges to: \[ S_\infty=\frac{a_1}{1-r}. \] If \(|r|\ge 1\), the series does not converge to a finite value.
Worked example: \(10+5+2.5+1.25+\cdots\)
Here \(a_1=10\) and \(r=\frac{1}{2}\). Since \(\left|\frac{1}{2}\right|<1\), the series converges.
\[ S_\infty=\frac{10}{1-\frac{1}{2}}=\frac{10}{\frac{1}{2}}=20. \]
Quick identification and setup
A sum is a geometric series if the ratio of consecutive terms is constant. For terms \(t_1,t_2,t_3,\dots\), compute \(t_2/t_1\) and \(t_3/t_2\). If the ratios match, that value is \(r\).
| Goal | What to extract | Formula to use |
|---|---|---|
| Finite sum of first \(n\) terms | \(a_1\), \(r\), \(n\) | \(S_n=a_1\cdot\frac{1-r^n}{1-r}\) for \(r\ne 1\) |
| Infinite sum | \(a_1\), \(r\) and check \(|r|<1\)< /td> | \(S_\infty=\frac{a_1}{1-r}\) (only if \(|r|<1\))< /td> |
Visualization: partial sums approaching the infinite sum
The diagram below uses the convergent geometric series \(10+5+2.5+1.25+\cdots\). Bars show the first several partial sums \(S_1,S_2,\dots,S_6\) approaching the limit \(S_\infty=20\).
Summary
A geometric series is a sum of terms with a constant ratio \(r\). The finite sum is \(S_n=a_1\cdot\frac{1-r^n}{1-r}\) for \(r\ne 1\), and when \(|r| 1\) the infinite geometric series converges to \(S_\infty=\frac{a_1}{1-r}\).