Geometric series

A geometric series adds the terms of a geometric progression:Its ordinary infinite sum is the limit of a sequence of partial sums $S_N={\sum }_{j=0}^{N-1}a{r}^j$, 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 andWhen $a\ne 0$, this condition on the modulus of $r$ is also necessary: for $|r|\ge 1$ the terms fail to tend to zero. The all-zero series $a=0$ converges for every $r$. The finite geometric series identity and $r^N\to 0$ give the result with almost no extra algebra. ^[ref]

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 $0,1,0,1,\dots$. 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 $0.\overline{27}$ isA geometric distribution counting failures before the first success normalizes by the same identity: for $0<q<1$, the probabilities $(1-q)q^n$ on $n=0,1,\dots$ 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.