= Geometric series
{title2=$\sum_{n=0}^{\infty}ar^n$}
{wiki}
= Geometric sum
{synonym}
A geometric series adds the terms of a <geometric progression>:
$$
a+ar+ar^2+\cdots=\sum_{n=0}^{\infty}ar^n.
$$
Its ordinary infinite sum is the <limit of a sequence> of <partial sums> $S_N=\sum_{j=0}^{N-1}ar^j$, rather than a new operation that adds infinitely many terms at once. For $a,r$ that are <real numbers> or <complex numbers>, if $|r|<1$, the series is <absolutely convergent> and
$$
\boxed{\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}.}
$$
When $a\ne0$, this condition on the <modulus> of $r$ is also necessary: for $|r|\ge1$ the terms fail to tend to zero. The all-zero series $a=0$ converges for every $r$. The <finite geometric series> identity and $r^N\to0$ give the result with almost no extra algebra. https://www.jirka.org/ra/html/sec_series.html[Lebl, Basic Analysis I, §2.5, Proposition 2.5.5 and Example 2.5.10]{ref}
\Image[/geometric-series-terms-and-partial-sums.png]
{title=Decaying terms, accumulating sums, and the failure of an undamped alternating series}
With $a=1$ and $r=1/2$, the <partial sums> approach $2$ from below. Changing the <common ratio> to $-1/2$ makes the <partial sums> approach $2/3$ from alternating sides: their increments still shrink. At $r=-1$, those increments never shrink, and the <partial sums> remain $0,1,0,1,\ldots$. Substituting $r=-1$ into $1/(1-r)$ does not supply the ordinary sum of that divergent series.
The formula is useful because many apparently different problems share a fixed multiplier. A <repeating decimal> such as $0.\overline{27}$ is
$$
\frac{27}{100}+\frac{27}{100^2}+\cdots
=\frac{27/100}{1-1/100}=\frac3{11}.
$$
A <geometric distribution> counting failures before the first success normalizes by the same identity: for $0<q<1$, the <probabilities> $(1-q)q^n$ on $n=0,1,\ldots$ add to one. The <comparison test for series> uses a bound by a decaying <geometric progression> to establish <absolute convergence> without an exact formula for the series being studied. The <iterated contraction gives a summable orbit> result uses precisely this idea for repeated applications of a map.
The <remainder of a geometric series> gives more than an eventual <limit>: it states exactly how much an approximation omits. A <common ratio> close to one still gives a <convergent series>, but can require many terms for a useful approximation.
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