Geometric series Definition and 182 Threads

In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series






1
2



+



1
4



+



1
8



+



1
16



+

⋯


{\displaystyle {\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots }
is geometric, because each successive term can be obtained by multiplying the previous term by 1/2. In general, a geometric series is written as a + ar + ar2 + ar3 + ... , where a is the coefficient of each term and r is the common ratio between adjacent terms. Geometric series are among the simplest examples of infinite series and can serve as a basic introduction to Taylor series and Fourier series. Geometric series had an important role in the early development of calculus, are used throughout mathematics, and have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance.
The distinction between a progression and a series is that a progression is a sequence, whereas a series is a sum.

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  1. A

    What is the Proof for the Sum to Infinity of an Infinite Geometric Progression?

    Homework Statement An infinite geometric progression is such that the sum of all the terms after the nth is equal to twice the nth term. Show that the sum to infinity of the whole progression is three times the first term. Homework Equations [/B] S_{n} = \frac{a(1-r^n)}{1-r}\\ S_{\infty} =...
  2. F

    Simple geometric series question

    Take the case for the mean: \bar{x} = \frac{1}{N} \Big( \sum_{i=1}^Ni \Big) If we simply use the formula for the sum of a geometric series, we get \bar{x} = \frac{N}{2} (2a + (N - 1)d) where a and d both equal 1, so we simply get the result \bar{x} = \frac{1}{2} (N + 1)...
  3. M

    Understanding the Formula for the Sum of a Geometric Series

    For the question, shouldn't the sum be a(1/1-r) since we know lrl < 1 then that rn → 0 as n → ∞? I just don't quite understand why they wrote the sum is a(r/1-r). Is there a specific reason they did this? This is just a regular geometric series right? Is there any difference since the sum starts...
  4. jdawg

    Making a geometric series that converges to a number

    Homework Statement How do you come up with a geometric series that converges to a number like 2? I'm kind of confused on how to work backwards through the problem. If someone could provide me with an example, that would be great! Homework Equations The Attempt at a Solution
  5. S

    Geometric series involving logarithms

    Homework Statement A geometric series has first term and common ratio both equal to ##a##, where ##a>1## Given that the sum of the first 12 terms is 28 times the sum of the first 6 terms, find the exact value of a. Hence, evaluate log_{3}(\frac{3}{2} a^{2}+ a^{4}+...+ a^{58}) Giving...
  6. S

    How Do You Derive the Common Ratio in a Geometric Series?

    Homework Statement In a geometric series, the first term is ##a## and the last term is ##l##, If the sum of all these terms is ##S##, show that the common ratio of the series is ##\frac{S-a}{S-l}##Homework Equations Sum of geometric seriesThe Attempt at a Solution I was thinking to use the sum...
  7. B

    Alternative Bound on a Double Geometric Series

    If |a_{mn}x_0^my_0^n| \leq M then a double power series f(x,y) = \sum a_{mn} x^m y^n can be 'bounded' by a dominant function of the form \phi(x,y) = \tfrac{M}{(1-\tfrac{x}{x_0})(1-\tfrac{y}{y_0})}, obviously derived from a geometric series argument. This is useful when proving that analytic...
  8. B

    Word Problem with Geometric Series

    Homework Statement The total reserves of a nonrenewable resource are 600 million tons. Annual consumption, currently 20 million tons per year, is expected to rise by 1% each year. After how many years will the reserve be exhausted? Part 2. Instead of Increasing by 1% each year, suppose...
  9. J

    Proper form for geometric series with a neg inside

    Homework Statement Curious about this ...I have to find the sum. Homework Equations The Attempt at a Solution Ʃ (1/4)(-1/3)^n from 1 to infinity I want to know the proper form and why. Is it (1/4) Ʃ (-1/3)(1/3)^n-1 or (1/4) Ʃ (1/3)(-1/3)^(n-1) You get different answers
  10. J

    Values for x which converge. (geometric series)

    Homework Statement Find the values of x for which the series converges. Find the sum of the series for those values of x. Homework Equations Sum (x+2)^n from n =1 to ∞ The Attempt at a Solution OK so this is maybe fudging it. I said that the argument x+2 will diverge if the...
  11. N

    MHB Why is my solution for part b) of the geometric series question incorrect?

    Please refer to the attached sheet. I need help with part b) for part a) I did: $\sum\limits_{n=0}^{\infty} a^n = \frac{1}{1-a}$ So for $\sum\limits_{x=0}^{\infty} a^{2x}$ $a^{2x} = (a^2)^x$ and $\sum\limits_{x=0}^{\infty} (a^2)^x = \frac{1}{1-a^2}$ for part b) the solutions say i am wrong...
  12. Lebombo

    Difference between 2 Sum of n terms of geometric series formulas

    Difference between 2 "Sum of n terms of geometric series" formulas Notation A) S_{n}= \sum_{k=0}^{n - 1} ar^{k} = ar^{0} + ar^{1} + ar^{2} +...+ ar^{n-1} = \frac{a(1-r^{n})}{1-r} Proof: S_{n}= ar^{0} + ar^{1} + ar^{2} +...+ ar^{n-1} - r*S_{n}= ar^{1} + ar^{2} + ar^{3} +...+ ar^{n}...
  13. M

    What is the sum of the two separate series in the original infinite series?

    Homework Statement Determine whether the serie is convergent or divergent , if it is convergent find its sum. Ʃ∞n=1 (1 + 2n )/ 3n Homework Equations Ʃa(r)n-1 = a / (1-r) r < 1 is converging or if r > 1 diverging The Attempt at a Solution Well I can see its a geometric series...
  14. B

    Why Is My Calculation of the Geometric Series Incorrect?

    \sum^{\infty}_{k=4} \frac{1}{5^{k}} ar^n=a/(1-r) Here is what I did: a=5 r=1/625 (1/5)/(1-1/625)=(1/5)/(624/625)=625/3120=125/624 The Answer is 1/500 Where am I going wrong?
  15. P

    Arithmetic and Geometric Series (tortoise and the hare)

    Homework Statement The question is attachedk Homework Equations Sn = n/2[2a+(n-1)d] Sn = (a x (1-r^n))/1-r The Attempt at a Solution I already found the general formulas: Tortoise: Sn = n/2(40) Hare: Sn = (1000 x [1-0.5^n])/0.5 And I know that there tortoise will finish the...
  16. K

    Geometric series with modified terms

    I am looking for a way to sum some numbers. I understand that if I want to sum pi, I can use the geometric series: \sum\limits_{i=0}^N p^{i} = \frac{1-p^{N+1}}{1-p} But can anyone help me with what to do when I need: \sum\limits_{i=0}^N p^{i} q^{ti} where t is just a constant...
  17. A

    MHB Formula for a geometric series with variable common ratio

    I have the following summation where a is a positive constant and can be > 1 $$ \sum_{k=2}^{n}a^{n-k} $$ I am trying to find a general formula for this summation, which turns out to be a geometric series with a as the common ratio. I have worked out the following: $$ \sum_{k=2}^{n}a^{n-k} =...
  18. L

    Comparison Test and Geometric Series

    The Problem: Let a(n) = (n^2)/(2^n) Prove that if n>=3, then: (a(n+1))/(a(n)) <= 8/9 By using this inequality for n = 3,4,5,..., prove that: a(n+3) <= ((8/9)^n)(a(3)) Using the comparison test and results concerning the convergence of the geometric series, show that: The...
  19. H

    Proof of 2nd Derivative of a Sum of a Geometric Series

    Homework Statement I am trying to prove how \(g''(r)=\sum\limits_{k=2}^\infty ak(k-1)r^{k-2}=0+0+2a+6ar+\cdots=\dfrac{2a}{(1-r)^3}=2a(1-r)^{-3}\). I don't know what I am doing wrong and am at my wits end. The Attempt at a Solution (The index of the summation is always k=2 to infinity)...
  20. B

    Geometric Series Test Rather Than Integral Test

    Homework Statement \sum_{n=1}^{\infty} \frac{1}{2^n} Homework Equations The Attempt at a Solution Could I some how manipulate this to fit a geometric series, so that I may instead use the geometric series test?
  21. B

    Simplifying finite geometric series expression

    Homework Statement I've come across the type of sum in several places/problems but seem to be making no progress in trying to simplifying it further. We have a finite series of some exponential function. \sum_{n=0}^{N}e^{-na} Where a is some constant, a quantum of energy or a...
  22. S

    Is the Series Ʃ (1 + 2^n) / 3^n Convergent and What is Its Sum?

    Homework Statement Determine whether the series is convergent or divergent. Find the sum if possible Ʃ 1+2^n / 3^n n=1 -> infinity Homework Equations a/1-r The Attempt at a Solution I split it up so that the equation is now: Ʃ (1/3^n) + (2/3)^n n=1 -> infinity Ʃ (1/3^n)...
  23. A

    Algorithms for infinite geometric series via long division?

    Algorithms for infinite geometric series via long division?? I can't seem to find any algorithms for this on the internet easily. If I have a function of the form f(x)=\frac{a}{x+b} there should be an algorithm I can use to find some terms of the corresponding series \sum...
  24. T

    Sum sequence of a geometric series

    Homework Statement A 'supa-ball' is dropped from a height of 1 metre onto a level table. It always rises to a height equal to 0.9 of the height from which it was dropped. How far does it travel in total until it stops bouncing? Homework Equations The Attempt at a Solution The...
  25. T

    What is the Limit of a Geometric Series with a Fractional Common Ratio?

    Homework Statement Evaluate the problem: \sum_{k=1}^\infty \frac{3^{(k-1)}}{4^{(k+1)}} Homework Equations \displaystyle\lim_{n\rightarrow\infty} S_n = \frac{a}{1-r} The Attempt at a Solution I know that the limit of the partial sequence is what i need to help solve this, but can't...
  26. D

    Sum of Geometric Series with cosine?

    Homework Statement With a series like: pi^(n/2)*cos(n*pi) How am I meant to approach this? Do I use the Squeeze Theorem? Homework Equations The Attempt at a Solution
  27. D

    How many bounces does it take for a ball to travel 1854.94320091 feet?

    Homework Statement A ball is dropped from a 100 feet and has a 90% bounce recovery. How many bounces does it take for the ball to travel 1854.94320091 feet? Homework Equations -None- The Attempt at a Solution I know the ratio is .9 and the 'a one' value is 180, so I plugged those values in...
  28. T

    Terms of a geometric series and arithmetic series, find common ratio

    Homework Statement Different numbers x, y and z are the first three terms of a geometric progression with common ratio r, and also the first, second and fourth terms of an arithmetic progression. a. Find the value of r. b. Find which term of the arithmetic progression will next be equal to...
  29. R

    Calculating credit card debt using a geometric series

    A man gets a credit card and buys something that charges exactly 800 dollars to the card. The APR on the card is 18 % compounded monthly, and the minimum payment is 15 dollars a month. What is the expression for A(n), the balance on the card after n months? (This should be a geometric series)...
  30. K

    Use geometric series to find the Laurent series

    Use geometric series to find the Laurent series for f (z) = z / (z - 1)(z - 2) in each annulus (a) Ann(1,0,1) (b) Ann(1,1,∞) Ann(a,r,R) a= center, r=smaller radius, R=larger radius Ann(1,0,1)=D(1,1)\{0} My attempt: f(z)= -1/(z-1) + 2/(z-2) geometric series: Σ[n=0 to inf] z^n - 1/2...
  31. F

    Find the Sum of The Geometric Series

    Homework Statement Ʃ(1 to infinity) (2/3)^(3n)Homework Equations For a geometric series, the series converges to a/1-r The Attempt at a Solution I'm really just confused about how to manipulate this so that it has a form of ar^n, especially since it starts at 1 rather than 0. I know that...
  32. B

    Proof without words: Geometric series

    On this website http://www.albany.edu/~bd445/Eco_466Y/Slides/Infinite_Geometric_Sum,_Proof_Without_Words.pdf there is a "proof without words" of the sum of the infinite geometric series. However, I don't understand what makes the proof valid. In what order were the constructions done etc.? What...
  33. M

    Arithmetric and Geometric Series

    Homework Statement A ) A company produces microchips. It has some in storage and produces 34 an hour. After 1 hour it has a total of 3428 microchips i) How many chips will the company have a week later, assuming the production continues 24 hours a day? ii)An order is put in for 13,526...
  34. S

    Divergence of the geometric Series at r=1

    Now for the proof of convergence/divergence of the geometric series we first deduce the Nth partial sum which is given by: \frac{r(1-r^n)}{1-r} Now for 0<r<1 this become \frac{1}{1-r} which clearly converges by AOL At r>1 it's similarly obvious why it diverges. But at r=1, I'm a bit...
  35. R

    MHB Taylor and Geometric Series questions

    I've spent all day on this problem and am wasting precious time needed for other work - please give any input you can! The question: given two wages, w1 and w2 where w2 > w1... a. the difference between the wages as a proportion of the lower: a = (w2 - w1) / w2 b. the difference between the...
  36. P

    MHB What is the sum to infinity of this nearly geometric series?

    Find the sum to infinity of the series $1 +2z +3z^2+4z^3+...$
  37. J

    Geometric series - positive and negative ratio

    Hello, Second term of a geometric series is 48 and the fourth term is 3... Show that one possible value for the common ratio, r, of the series is -1/4 and state the other value. ar=48, ar^3= 3... so ar^3/ar=3/48 which simplifies to r^2 = 1/16, therefore r = 1/4 Can anyone explain where...
  38. M

    Closed Form for nth Partial Sum of a Geometric Series

    Homework Statement Find a closed form for the nth partial sum, and determine whether the series converges by calculating the limit of the nth partial sum. 1. 2+2/5 + 2/25+...2/5k-1 Homework Equations The Attempt at a Solution What I did was I found out it was a geometric...
  39. A

    Can Integrating a Generalized Geometric Series Reveal New Insights into √π?

    I just sent some time dicking around with the MacLaurin expansion of exp(-z2) to derive a series expression for √π, by integrating term-by-term along the real line. I'm not really concerned with wether this is a useful or well-studied expression, I just thought it would be a fun exercise...
  40. A

    Sum of a geometric series up to infinity

    Homework Statement A geometric series had first term 54 and 4th term 2. (i) What is the common ratio? (ii) Find the sum to infinity of the series. (iii) After how many terms is the sum of the series greater than 99% of the sum to infinity? Homework Equations N/A The Attempt at a...
  41. S

    Convergence/Divergence of a Geometric series with a Factorial

    Homework Statement Determine if the sequence {an} below converges or diverges. Find the limit of each convergent sequence an = n!/nn Hint: Compare with 1/n . Find the limit of the sequence {an} if it converges. I missed the lesson on factorials, and the book is useless. Sorry...
  42. S

    Trying to find interval of convergence for a geometric series

    Homework Statement here is the series: \sum^{\infty}_{n=0}x(-15(x^{2}))^{n} Homework Equations The Attempt at a Solution I know that -1<-15x^{2}<1 for convergence (because of geometric series properties) but I run into a problem here: -1/15<x^{2}<1/15 You can't...
  43. J

    Any way to figure out what this finite geometric series sums to?

    I would like to find a nice formula for \sum_{k=0}^{n - 1}ar^{4k}. I know that \sum_{k=0}^{n - 1}ar^{k} = a\frac{1 - r^n}{1 - r} and was wondering if there was some sort of analogue.
  44. H

    How do I sum a geometric series with complex numbers?

    Homework Statement Homework Equations a(1-r^[n+1])/(1-r) The Attempt at a Solution So I wrote it as e^(-iNz) [1 + e^(iz) + e^(2iz) + ... + e^(2iNz)] Let r = e^(iz), a=e^(-iNz) a [1 + r + r^2 + ... + r^(2N)] From here I'm not sure what to do. I tried letting n=2N, and...
  45. M

    What is the Sum of a Geometric Series with a Given Initial Value and Ratio?

    Homework Statement I already counted V_{0}=-1 and q=\frac{1}{3} given: V_{n}=1-\frac{2}{U_{n}} Homework Equations count: \sum_{k=0}^{n}V_{k} The Attempt at a Solution i counted the sum and i got : ((\frac{1}{3})^{n+1}-1)(\frac{2}{3}) is that correct?
  46. C

    Limits, geometric series, cauchy, proof HELP

    i guys, I'm stuck on wording of a homework assignment and thought you might be able to help me. There are several questions... Consider the geometric series: (Sum from k=0 to infinity) of ar^k and consider the repeating decimal .717171717171 for these problems: Question 1: Find a formula...
  47. Z

    How Many Mixes to Complete Geometric Dilution of Two Powders?

    1. Homework Statement you are geometrically diluting/mixing 0.1 g of powder A with 100g of powder B, how many times do you have to mix the 2 together to finish the process? *each time you can only mix an equal portion of powder B to what you currently have mixed. Eg. 1:1 2:2 4:4 3...
  48. X

    Simplifying to a Geometric Series

    Homework Statement I have a question with asks to solve a differential equation via power series and I've done everything up to finding the recurrence relation which is a_{n+2} = -\frac{a_{n}}{n+2} Given the initial conditions a_{o} = 1 and a_{1} = 0 I'm trying to simplify the series into a...
  49. M

    Convergence Proof (As Part of Geometric Series Sum)

    Homework Statement I am trying to prove the sum of a geometric series, but one of the steps involves deriving this result: \lim_{n\to\infty}r^{n}=0 so that you can simplify the sum of a geometric series, where I have got to this stage: S_{\infty} = \frac{a(1-r^{\infty})}{1-r}...
  50. Rapier

    Sum of Geometric Series: Ʃ(3→∞) 3(.4)^(n+2)

    Homework Statement Find the sum of Ʃ(3→∞) 3(.4)^(n+2) Homework Equations Sum of Geometric Series = ao/(1 - r), ao=3, r = .4 = 2/5 The Attempt at a Solution I thought that I could use the definition of the sum of a geometric series (above) to determine the sum of this equation...
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