Geometric Series
Sequences of lengths occur naturally, and with notable frequency they fall into the
category of the
Geometric 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.
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
- a summation symbol, $\sum$, Greek capital sigma letter,
- constant numbers (in the coefficient, base, and/or exponent),
- one or more appearances of the variable being summed over,
signified in the subscript of the $\sum$-symbol,
conventionally written as $n$.
Naturally, with familiarity, one associates the place of the variable with the name of the series,
e.g. the Geometric Series is associated with one appearance of $n$ and it's in the exponent position.
This is a diagram of the geometric series for a base of 1/2.
This is a diagram of the geometric series for a base of 1/3.
The Wikipedia link has a picture of this drawn with contiguous squares inscribed in
a unit square (a square with a side length equal to one).
That sequence of sides ($1/2 + 1/4 + 1/8 +...$) clearly sums 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).
The same argument doesn't hold for any other $r$-value because they all converge at different rates,
e.g. $r=1/3$ converges to $3/2$ (that's $1+1/2$ for a series that starts with
$1=r^0$
)
, and it's not obvious without the algebra for the formula.
The formula for the geometric series is
$$
S_{m}=\sum^{m}_{x=0} r^x = \frac{1 - r^{m+1}}{1-r}
$$
where $r$ is a constant fraction $0\lt r \lt 1$.
This compact formula is apparent after the following steps.
First we take the partial series (sum of the series to some finite $n$),
and multiply by the binomial $(1-r)$, term by term.
$$
(1-r)\left(\sum^{m}_{x=0} r^x\right)
$$
Where we have multiplied the Geometric series by $(1-r)$.
Next we subtract, as an intermediate step, $r$-times the series from the copy of the series ($(1-r)S_m=(1)(S_m)-(r)(S_m)$).
$$
= (1 +r + r^2 +\dots+r^m)-(r + r^2 + r^3 + \dots+r^{m+1})
$$
Since there isn't a one in the second set of paretheses (series on the right), and there is one on the left, we have a one in the total, and so on for the $r^1$ terms, which each cancel.
$$
= (1) +(r-r) + (r^2-r^2) +\dots+(r^m-r^m) +r^{m+1})
$$
Finally we are left with the following reduction:
$$
= 1-r^{m+1}
$$
And so the formula is completed.
[Arfken]
Changing the variable in the Geometric series formula, as $m=n-1$,
shifts the formula by one, which is an equivalent statement:
$$
S_{n-1}=\sum^{n-1}_{x=0} r^x = \frac{1 - r^n}{1-r}
$$
Note the series includes the $r^0=1$ term,
not in the geometric Wikipedia diagram,
so if you want to know what the formula is for the sum of sides of the inscribed squares,
then just subtract one
(because all the subsequent members of the infinite sequences are the same).
Below are plots of the geometric series function for $r=\{1/4, 1/3, 1/2, 3/5\}$,
which includes the points continuously between each of the integer points of $x$.
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 1/3.
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 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 $r<1$, and as
$r$ gets closer to one the function gets arbitrarily large,
because you can make the denominator, $1-r$, arbitrarily small.
