Editing Geometric series (section)

== Convergence of the series and its proof ==
The convergence of the infinite sequence of partial sums of the infinite geometric series depends on the [[Magnitude (mathematics)|magnitude]] of the common ratio <math>r</math> alone:
* If <math>\vert r \vert < 1</math>, the terms of the series approach zero (becoming smaller and smaller in magnitude) and the sequence of partial sums <math>S_n</math> converge to a limit value of <math display="inline">\frac{a}{1-r}</math>.{{r|vpr}}
* If <math>\vert r \vert > 1</math>, 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 [[Divergent series|diverges]].{{r|vpr}} 
* If <math>\vert r \vert = 1</math>, 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 <math>r=1</math>, all the terms of the series are the same and the <math>|S_n|</math> grow to infinity. When <math>r = -1</math>, the terms take two values <math>a</math> and <math>-a</math> alternately, and therefore the sequence of partial sums of the terms [[Oscillation (mathematics)|oscillates]] between the two values <math>a</math> and 0. One example can be found in [[Grandi's series]]. When <math>r=i</math> and <math>a = 1</math>, the partial sums circulate periodically among the values <math>1, 1 + i, i, 0, 1, 1+ i,i,0, \ldots</math>, never converging to a limit. Generally when <math display="block">r= e^\frac{2\pi i}{\tau}</math> for any integer <math>\tau</math> and with any <math>a \neq 0</math>, the partial sums of the series will circulate indefinitely with a period of <math>\tau</math>, never converging to a limit.{{r|apostol}}

The [[rate of convergence]] shows how the sequence quickly approaches its limit. In the case of the geometric series&mdash;the relevant sequence is <math> S_n </math> and its limit is <math> S </math>&mdash;the rate and order are found via
<math display="block"> \lim _{n \rightarrow \infty} \frac{\left|S_{n+1} - S\right|}{\left|S_{n}-S\right|^{q}}, </math>
where <math> q </math> represents the order of convergence. Using <math display="inline"> |S_n - S| = \left| \frac{ar^{n+1}}{1-r} \right| </math> and choosing the order of convergence <math> q = 1 </math> gives:{{r|nw}}
<math display="block"> \lim _{n \rightarrow \infty} \frac{\left| \frac{ar^{n+2}}{1-r} \right|}{\left| \frac{ar^{n+1}}{1-r} \right|^{1}} = |r|.</math>
When the series converges, the rate of convergence gets slower as <math>|r|</math> approaches <math>1</math>.{{r|nw}} The pattern of convergence also depends on the [[Sign (mathematics)|sign]] or [[Argument (complex analysis)|complex argument]] of the common ratio. If <math>r > 0</math> and <math>|r| < 1</math> then terms all share the same sign and the partial sums of the terms approach their eventual limit [[Monotonic sequence|monotonically]]. If <math>r < 0</math> and <math>|r| < 1</math>, adjacent terms in the geometric series alternate between positive and negative, and the partial sums <math> S_n </math> of the terms oscillate above and below their eventual limit <math>S</math>. For complex <math>r</math> and <math>|r| < 1,</math> the <math>S_n</math> converge in a spiraling pattern.

The convergence is proved as follows. The partial sum of the first <math>n + 1</math> terms of a geometric series, up to and including the <math>r^{n}</math> term,
<math display="block"> S_n = ar^0 + ar^1 + \cdots + ar^{n} = \sum_{k=0}^{n} ar^k, </math>
is given by the closed form
<math display="block"> S_n = \begin{cases}
 a(n + 1)  & r = 1\\
 a\left(\frac{1-r^{n+1}}{1-r}\right) & \text{otherwise}
\end{cases} </math>
where <math> r </math> is the common ratio. The case <math>r = 1</math> is merely a simple addition, a case of an [[arithmetic series]]. The formula for the partial sums <math>S_n</math> with <math>r \neq 1</math> can be derived as follows:{{sfnp|Apostol|1967|pp=388–390}}{{r|as|pm}}
<math display="block">\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}</math>
for <math> r \neq 1 </math>. As <math>r</math> approaches 1, polynomial division or [[L'Hôpital's rule]] recovers the case <math>S_n = a(n + 1)</math>.{{sfnp|Apostol|1967|pp=292–295}} 

[[File:Geometric_progression_sum_visual_proof.svg|thumb|[[Proof without words]] of the formula for the sum of a geometric series if <math> |r| < 1 </math> and <math> n \to \infty </math>, the <math> r^n </math> term vanishes, leaving <math display="inline"> \lim_{n \to \infty} S_n = \frac{a}{1-r} </math>. This figure uses a slightly different convention for <math> S_n </math> than the main text, shifted by one term.]]
As <math>n</math> approaches infinity, the absolute value of {{math|''r''}} must be less than one for this sequence of partial sums to converge to a limit. When it does, the series [[Absolute convergence|converges absolutely]]. The infinite series then becomes
<math display="block"> \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}
</math>
for <math> |r| < 1 </math>.{{sfnp|Apostol|1967|pp=388–390}}

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.{{sfnp|Apostol|1967|pp=399–400}}