Inmathematics, aninfinite geometric series of the form

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}a{r}^{n-1}=a+ar+a{r}^2+a{r}^3+\cdots }$

isdivergent if and only if${\displaystyle |r|>1.}$ Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric series to a sum that agrees with the formula for the convergent case

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}a{r}^{n-1}=\frac{a}{1-r}.}$

This is true of any summation method that possesses the properties ofregularity, linearity, and stability.

In increasing order of difficulty to sum:

• 1 − 1 + 1 − 1 + ⋯, whose common ratio is−1
• 1 − 2 + 4 − 8 + ⋯, whose common ratio is −2
• 1 + 2 + 4 + 8 + ⋯, whose common ratio is2
• 1 + 1 + 1 + 1 + ⋯, whose common ratio is 1.

It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-calledBorel-Okada principle: If aregular summation method assigns$\sum _{n=0}^{\mathrm{\infty }}{z}^n$ to${\displaystyle 1/(1-z)}$ for all$z$ in a subset${\displaystyle S}$ of thecomplex plane, given certain restrictions on${\displaystyle S}$, then the method also gives theanalytic continuation of any other function$f(z)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{z}^n$ on the intersection of${\displaystyle S}$ with theMittag-Leffler star for${\displaystyle f(z)}$.^[1]

Ordinary summation succeeds only for common ratios

• Cesàro summation
• Abel summation

• Euler summation

The series isBorel summable for everyz with real part < 1.

Certainmoment constant methods besides Borel summation can sum the geometric series on the entire Mittag-Leffler star of the function 1/(1 −z), that is, for allz except the rayz ≥ 1.^[2]

• Divergent series
• Geometric series