When there are infinitely many terms in a geometric progression, the geometric series is: a + ar + ar^2 + ar^3 + \cdots = \sum_{k=0}^\infty ar^k. The result of an infinite series can be either
convergent or
divergent. Convergence means there is a value after summing infinitely many terms, whereas divergence means no value after summing. The convergence of a geometric series can be described depending on the value of a common ratio. For instance,
Grandi's series is divergent geometric series that can be expressed as 1 - 1 + 1 - 1 + \cdots , where the initial term is 1 and the common ratio is -1 ; this is because it has three different values.
Decimal numbers that have
repeated patterns that continue forever can be interpreted as geometric series and thereby converted to expressions of the
ratio of two integers. For example, the repeated decimal fraction 0.7777\ldots can be written as the geometric series 0.7777\ldots = \frac{7}{10} + \frac{7}{10} \left(\frac{1}{10}\right) + \frac{7}{10} \left(\frac{1}{10^2}\right) + \frac{7}{10} \left(\frac{1}{10^3}\right) + \cdots,where the initial term is a = \tfrac{7}{10} and the common ratio is r = \tfrac{1}{10}. To determine its convergence, one may look up the
magnitude of the common ratio r alone: • If \vert r \vert , the terms of the series approach zero (becoming smaller and smaller in magnitude) and the sequence of partial sums S_n converge to a limit value of \frac{a}{1-r}. • If \vert r \vert > 1, the terms of the series become larger and larger in magnitude and the partial sums of the terms also get larger and larger in magnitude, so the series
diverges. • If \vert r \vert = 1, the terms of the series become no larger or smaller in magnitude and the sequence of partial sums of the series does not converge. When r=1, all the terms of the series are the same and the |S_n| grow to infinity. When r = -1, the terms take two values a and -a alternately, and therefore the sequence of partial sums of the terms
oscillates between the two values a and 0. One example can be found in
Grandi's series. When the common ratio is the
imaginary unit r=i and a = 1, the partial sums circulate periodically among the
complex numbers , , , , , , , , ..., never converging to a limit. When the common ratio is a
root of unity r= e^{2\pi i p / q} for a rational number p/q in
lowest terms and with any a \neq 0, the partial sums of the series will circulate indefinitely with a period of q, never converging to a limit. The
rate of convergence shows how the sequence quickly approaches its limit. In the case of the geometric series—the relevant sequence is S_n and its limit is S —the rate and order are found via \lim _{n \rightarrow \infty} \frac{\left|S_{n+1} - S\right|}{\left|S_{n}-S\right|^{q}}, where q represents the order of convergence. Using |S_n - S| = \left| \frac{ar^{n+1}}{1-r} \right| and choosing the order of convergence q = 1 gives: \lim _{n \rightarrow \infty} \frac{\left| \frac{ar^{n+2}}{1-r} \right|}{\left| \frac{ar^{n+1}}{1-r} \right|^{1}} = |r|. When the series converges, the rate of convergence gets slower as |r| approaches 1. The pattern of convergence also depends on the
sign or
complex argument of the common ratio. If r > 0 and |r| then terms all share the same sign and the partial sums of the terms approach their eventual limit
monotonically. If r and |r| , adjacent terms in the geometric series alternate between positive and negative, and the partial sums S_n of the terms oscillate above and below their eventual limit S. For complex r and |r| the S_n converge in a spiraling pattern.
Proof of convergence The convergence is proved as follows. The partial sum of the first n + 1 terms of a geometric series, up to and including the r^{n} term, S_n = ar^0 + ar^1 + \cdots + ar^{n} = \sum_{k=0}^{n} ar^k, is given by the closed form S_n = \begin{cases} a(n + 1) & r = 1\\ a\left(\frac{1-r^{n+1}}{1-r}\right) & \text{otherwise} \end{cases} where r is the common ratio. The case r = 1 is merely a simple addition, a case of an
arithmetic series. The formula for the partial sums S_n with r \neq 1 can be derived as follows: \begin{align} S_n &= ar^0 + ar^1 + \cdots + ar^{n},\\ rS_n &= ar^1 + ar^2 + \cdots + ar^{n+1},\\ S_n - rS_n &= ar^0 - ar^{n+1},\\ S_n\left(1-r\right) &= a\left(1-r^{n+1}\right),\\ S_n &= a\left(\frac{1-r^{n+1}}{1-r}\right), \end{align} for r \neq 1 . As r approaches 1, polynomial division or
L'Hôpital's rule recovers the case S_n = a(n + 1). of the formula for the sum of a geometric series if |r| and n \to \infty , the r^n term vanishes, leaving \lim_{n \to \infty} S_n = \frac{a}{1-r} . This figure uses a slightly different convention for S_n than the main text, shifted by one term. As n approaches infinity, the absolute value of must be less than one for this sequence of partial sums to converge to a limit. When it does, the series
converges absolutely. The infinite series then becomes \begin{align} S &= a+ar+ar^2+ar^3+ar^4+\cdots\\ &= \lim_{n \rightarrow \infty} S_n\\ &= \lim_{n \rightarrow \infty} \frac{a(1-r^{n+1})}{1-r} \\ &= \frac{a}{1-r} - \frac{a}{1-r} \lim_{n \rightarrow \infty} r^{n+1} \\ &= \frac{a}{1-r}, \end{align} for |r| . This convergence result is widely applied to prove the convergence of other series as well, whenever those series's terms can be bounded from above by a suitable geometric series; that proof strategy is the basis for the
ratio test and
root test for the convergence of infinite series. == Connection to the power series ==