Geometric Series

Sequences of lengths occur naturally, and with notable frequency they fall into the category of the Geometric Series. The series of ratios expressable by a compact formula, being considered first as a finite series of ratios and then taken a step further to the case when the number of ratios being summed goes to infinity—which is daunting at first glance, algebra is needed to reveal the closed form for the series.

The series for a common ratio of $1/2=0.5$ is solvable geometrically by drawing the sequence, as the Wikipedia page on the subject displays, which doesn't itself hint at the algrebra required to discover the generalized formula. For each set of sequential additive (and subtractive) member terms (a.k.a. an infinite series), there is the use of a parameter (variable) which shows how the sequence is constructed from the following notation elements:

1. a summation symbol, $\sum$, Greek capital sigma letter, examples:
   1. $\sum\limits_0^3 n = 0 + 1 + 2 + 3 = 6 $, where the formula under summation expands as a series of terms,
2. constant numbers (in the coefficient, base, and/or exponent),
   1. $\sum\limits_{1}^4 n \pi = (1 + 2 + 3 + 4)\pi = 10\pi $, where $\pi$ can be any number
   2. $\sum\limits_{-1}^1 n^2 = (-1)^2 + 0^2 + 1^2 = 2 $, where the exponent is constant
3. one or more appearances of the variable being summed over, signified in the subscript of the $\sum$-symbol, conventionally written as $n$.
   1. $\sum\limits_{n=1}^3 (n-\frac{3}{2})(n)=(-\frac{1}{2}) + 1 + \frac{9}{2} = 5$

The geometric Series is associated with one appearance of $n$ and it's in the exponent position.

The geometric series is expressed as a sum of terms, each of which has gone from kernel plus indexical substitution to the term. The kernel is, $r^n: 0 \lt r \lt 1$, where the index, $n$, starts with zero and goes to infinity.

\begin{equation}\label{GS_S_g} S_g=\sum\limits_{n=0}^\infty\,r^n = r^0 + r^1 + r^2 + r^3 + \cdots \end{equation}

Since the starting term is $n=0$, then the series is simply offset by one regardless of $r$ ( $1=r^0$ ) so it is omitted for brevity (the $n\gt0$ terms are what's important), it was used that way with the the inscription into the unit, and used that way in the following.

For the $r=0.5$ geometric series, we observe the post-zeroth sequence displayed (each on top the previous) in a unit-length horizontal rectangle as forever adding a half again of the previous term in the summation thus quite probably equalling one (we will see the formula below), by induction on the following seven terms of the series:

This is a diagram of the geometric series minus the zeroth term, for a base ratio, ($r$), of $\frac{1}{2}$, going up to the $2^{-7}$, $128^{-1}$, or ("$n=7$") -term.

When we employ an $r$ value of less than one half then the summation is less than one (total sum less than two, including the zeroth term), which is graphically depicted in the same unit rectangle, so we generalize $r\lt 1/2$ for any $r$ the sum gets closer to zero, but we really need a formula.

This is a diagram of the geometric series for a base ($r$) of $1/3$, going up to the $3^{-7}$, $2187^{-1}$, or ("$n=7$") -term.

The Wikipedia link above has a picture of this drawn with contiguous (the next beginning where the previous ended) squares inscribed in a unit square (a square with a side length equal to one). That sequence of sides ($1/2 + 1/4 + 1/8 +\ldots$) correctly (appears to) sum to one, because the sequence is comprised of steps which each leave a remainder of the unit (1) as $2^{-n}$—halving the distance to the unit, thus never reaching it in an infinitely precise way, and at the same time the partial-sum G.S. up to $n=3$ amounts to $0.875$, leaving a maximum of $0.125$ for the remaining terms in the infinite series sum giving appearance that the $r=0.5$ series summing to one. $r=1/3$ converges to $1/2$, which really is not obvious without the algebra for the formula. The formula for the geometric series is as follows:

Where $r$ is a constant fraction $0\lt r \lt 1$, with the variable being $m$. This compact formula is apparent after the following steps, thanks to Bernouli. First we take the partial series (sum of the series to some finite $m$), and multiply by the binomial $(1-r)$:

$$ (1-r)\left(\sum^{m}_{n=0} r^n\right) $$

Note the series we want to work with is the one that starts with $n=0$. Which is to say subtract, $r$-times the zero-based series from that same series, like so: $(1-r)S_m=(1)(S_m)-(r)(S_m)$.

$$ = (1 +r + r^2 +\dots+r^m)-(r + r^2 + r^3 + \dots+r^{m+1}) $$

Since there isn't a one in the second set of parentheses (series on the right starts with the $n=1$ term), and there is a $n=0$ term in the left parentheses, we have a one in the sum output followed by the last term left unpaired in negation $rr^m=r^{m+1}$.

$$ = (1) +(r-r) + (r^2-r^2) +\dots+(r^m-r^m) + (0-r^{m+1}) $$

Finally we are left with the following reduction:

$$ = 1-r^{m+1} $$

And so the formula is completed, after adjusting the formula for the appropriate starting term of the desired series. Changing the variable in the geometric series formula, as $m=n-1$, shifts the formula by one on the input:

$$ S_{n-1}=\sum^{n-1}_{x=0} r^x = \frac{1 - r^n}{1-r} $$

And now we can resolve the mystery of whether or not the case of $r=1/3$, excluding the zeroth term, does converge to $0.5$, which it does since $S_g(1/3)=3/2$. And, a look at the ratio of a half, $S_g(0.5)=2$, proving the infinite sum not including the zeroth term is one.

Below are plots of the partial-sums of the geometric series for $r=\{1/4, 1/3, 1/2, 3/5\}$, which includes the points continuously between each of the integer points ($n$) by using a continuous variable $x$ and highlights drawn for $n=x=\{1, 2, 5\}$. Note that the plot ordinate range starts at one because the $n=0$ term is being included.

This is a plot of the geometric series partial sum for a common ratio of 1/4.

This is a plot of the geometric series partial sum for a common ratio of 1/3.

This is a plot of the geometric series partial sum for a common ratio of 1/2.

This is a plot of the geometric series partial sum for a common ratio of 3/5.

Plot of the geometric series infinite sum over a range, as a function of the variable $r$, of common ratios $[0.1,0.7]$.

Plot of the geometric series infinite sum over a range, as a function of the variable $r$, of common ratios $[0.7,0.999]$.

The one-over-one-minus-$r$ function (the infinite sum formula) is valid for ratios $r\lt 1$, and as $r$ gets closer to one the function gets arbitrarily large, because you can make the denominator, $1-r$, arbitrarily small.