From 637d1966921b7fc8af239ce4364c8e7ba727547f Mon Sep 17 00:00:00 2001 From: Alexander Molnar <37451552+BTmathic@users.noreply.github.com> Date: Thu, 21 Mar 2019 09:06:58 -0400 Subject: [PATCH] Added content to stub (#30728) Provided definition, examples, as well as applications for their use and importance. --- .../mathematics/geometric-series/index.md | 160 +++++++++++++++--- 1 file changed, 132 insertions(+), 28 deletions(-) diff --git a/guide/english/mathematics/geometric-series/index.md b/guide/english/mathematics/geometric-series/index.md index f2209395ef..f49694ac52 100644 --- a/guide/english/mathematics/geometric-series/index.md +++ b/guide/english/mathematics/geometric-series/index.md @@ -3,42 +3,146 @@ title: Geometric Series --- ## Geometric Series +A geometric series is a sum of numbers, such as +``` +1 + 1/2 + 1/4 + 1/8 + 1/16 + ... +``` +where the first term is not zero and each consecutive pair of terms in the sum has the same ratio, called the *common ratio*. In the example above we have `1/(1/2) = 2`, `(1/2)/(1/4) = 2`, etc... While seemingly fairly simple this type of sum is found extensively in many areas including [geometry of real numbers](https://en.wikipedia.org/wiki/Cantor_set), [fractals](https://en.wikipedia.org/wiki/Geometric_series#Fractal_geometry), [probability](https://math.stackexchange.com/questions/1164163/introduction-to-probability-dice), [economics](https://en.wikipedia.org/wiki/Annuity#Proof_of_annuity-immediate_formula), converting rational numbers from decimal form to fraction form (see below) and more. -A geometric progression is a sequence of numbers, whose first term is non zero and each term is obtained by multiplying its proceding term by a constant quantity. This constant quantity is called common ratio of the GP. +We will show how to sum a geometric series when possible, give the formulas and then give some examples of this below. -The sum of this progression is called geometric series. +It is common to see the common ratio between terms denoted by `r`, so if a geometric series starts with first term `a`, the series is +``` +a + a*r + a*r^2 + a*r^3 + a*r^4 + ... +``` +When summing an infinite number of terms (such as 1 + 1 + 1 + 1 + ...) it is not clear the sum should be a number, but a remarkable property about geometric series is that the sum is always a number when `-1 < r < 1` and otherwise the sum is not a number (e.g., our `1 + 1 + 1 + ...` series tends to infinite, so is not a number). To see why, suppose we are only interested in the first `n + 1` terms of the series, +``` +a + a*r + a*r^2 + ... + a*r^n +``` +This is a sum of a `n + 1` numbers, so is certainly a real number itself. If we call this number `S_n`, so +``` +S_0 = a, +S_1 = a + a*r, +S_2 = a + a*r + a*r^2, +etc... +``` +we have +``` +S_n = a + a*r + a*r^2 + ... + a*r^n +``` +and multiplying both sides by `r` gives +``` +S_n*r = a*r + a*r^2 + a*r^3 + ... + a*r^n + a*r^(n+1) +``` +which looks almost like our original sum `S_n` on the right. We have an extra `a*r^(n+1)` and we are missing the first term `a`, so writing +``` +S_n*r = 0 + a*r + a*r^2 + a*r^n + a*r^(n+1) + = (a - a) + a*r + a*r^2 + ... + a*r^n + a*r^(n+1) + = a + a*r + a*r^2 + ... + a*r^n + a*r^(n+1) - a +``` +we can now see `S_n` on the right hand side again. Indeed, we see that +``` +S_n*r = S_n + a*r^(n+1) - a +``` +and +``` +a - a*r(n+1) = S_n - S_n*r = S_n*(1 - r) +``` +so that +``` +S_n = [a - a*r^(n+1)]/(1 - r) = a*[1 - r^(n+1)]/(1 - r). +``` +when `r` is not equal to 1 (we cannot divide by 0!). Now if `-1 < r < 1` then we know that `r^(n+1)` tends to 0 as `n` tends to infinity, so our sum `S_n` tends to +``` +S = a*[1 - 0]/(1 - r) = a/(1 - r). +``` +On the other hand, when `r > 1` or `r < 1` we know that `r^(n_1)` does not tend to a finite number, so the `S_n` do not tend to a finite number either. The only case we have not mentioned yet is when `r = 1`, but then the series is +``` +S_n = a + a*1 + a*1^2 + ... + a*1^n = a + a + ... + a = (n + 1)*a +``` +which certainly does not equal a number when `n` tends to infinity. -We can say if a1,a2,a3,....,an are in GP then +### Formula for geometric series - +If we have some real number `a` and some `r` such that `-1 < r < 1`, then the *finite* geometric series +``` +a + a*r + a*r^2 + ... + a*r^(n+1) +``` +has the sum +``` +a*[1 - r^(n+1)]/(1 - r) +``` +while the *infinite* geometric series +``` +a + a*r + a*r^2 + ... +``` +has the sum +``` +a/(1 - r). +``` - is known as common ratio of GP. - - if a is the first term then GP can be written as +### Examples -example of a GP : 2,4,8,16,.... +For example, the series with `a = 1` and `r = 1/2` above has sum +``` +1/(1 - 1/2) = 1/(1/2) = 2. +``` +(This particular sum is well known for [one](https://www.reddit.com/r/Jokes/comments/1423a4/an_infinite_number_of_mathematicians_walk_into_a/) or [two](https://www.reddit.com/r/Jokes/comments/929r9g/an_infinite_number_of_mathematicians_walk_into_a/) jokes.) -## The nth term of GP (Also known as General term) +The geometric series with first term `a = 2.5` and common ratio `r = -1/3` has a sum of +``` +2.5/[1 - (-1/3)] = 2.5/[1 + 1/3] = (5/2)/(4/3) = (5/2)*(3/4) = 15/8 +``` +whereas if we started with `a = -2.5` instead, the series has a sum of +``` +-2.5/[1 - (-1/3)] = -15/8. +``` -Let a be the first term, r be the common ratio and l be the last term of a GP, then nth term is given by +### Application for converting decimals to fractions - - - where - - # The sum of n terms of a GP - - Let a be the first term, r be the common ratio and l be the last term of a GP, then sum of n terms is given by: - - - and is not defined for r=1 +To see how this can be useful converting decimal numbers to fractions, consider the number `0.1111111...` with 1s repeating. As +``` +0.1 = 1/10, +0.01 = 1/100, +0.001 = 1/1000, +etc... +``` +we can rewrite this as +``` +0.11111... = 1/10 + 1/100 + 1/1000 + ... +``` +which is a geometric sum with first term `a = 1/10` and common ratio `r = 1/10`, so using our formula for the sum above, +``` +0.111... = (1/10)/(1 - 1/10) = (1/10)/(9/10) = (1/10)*(10/9) = 1/9. +``` -# Geometric Mean - -if we insert geometric mean between two numbers and , then +For a more complicated decimal, consider `0.124312431243...` where the repeating part is 1243. The decimal is equivalent to +``` +1243/10000 + 1243/10000^2 + 1243/10000^3 + ... +``` +and so our simple sum formula gives +``` +0.124312431243... = (1243/10000)/(1 - 1/10000) + = (1243/10000)*(10000/9999) + = 1243/9999 = 113/909 +``` -Geometric mean = - - - - +For the most complicated type of repeating decimal, consider one that starts with something different from the repeating pattern, e.g., `0.42567676767...` where after the initial 425 we repeat with 67 forever. This is simply +``` +0.425 + 0.000676767... +``` +and so we have +``` +0.425676767... = 425/1000 + 67/100000 + 67/(100000*100) + 67/(100000*100^2) + ... + = 425/1000 + (67/100000)/(1 - 1/100) + = 425/1000 + 67/99000) + = (425*99 + 67)/99000 = 21071/49500. +``` +This may be less pleasant without a calculator, but just as simple as the examples above with our wonderful formula! + +This also gives a nice validation that `0.999... = 1`. Indeed, we have +``` +0.999 = 9/10 + 9/100 + 9/1000 + ... + = (9/10)/(1 - 1/10) + = (9/10)/(9/10) = 1. +``` \ No newline at end of file