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. Y

    Can you simplify this geometric series for computing the z-transform?

    Hi, I have a problem with computing this geometric series. Homework Statement I have to compute \sum_{i=0}^\infty{(\frac{1}{2z})^{2k}} + \sum_{i=0}^\infty{(\frac{1}{3z})^{2k+1}}. It's for computing the z-transform of f[k]=0 for k<0 f[k]=(\frac{1}{2})^k for k=0,2,4,6...
  2. M

    Sum of infinite geometric series

    Homework Statement Homework Equations The Attempt at a Solution I don't get what I am doing wrong here, I have attached my solution below. The solution manual have their answer as 3e/(3-e). Thanks!
  3. S

    Coming up with & summing a geometric series

    Homework Statement a) Two friends, Jon and Bob, are sharing a loaf of bread. Jon eats half of the loaf, then Bob eats half of what remains, then Jon eats half of what remains and so on. How much of the loaf did each of them eat? b)Jon is hungrier and eats 2/3 of the loaf, then Bob eats half...
  4. C

    Math 30 pure geometric series

    helo this is a homework problem i got in math 30 pure i got an answer but i would like to know how to get it by using a formula? The exercise gose like this: Initially, a pendulum swings through an arc of 2feet. On each successive swing,the length of the arc is 0.9 of the previous length...
  5. cepheid

    Finite Sum - Modified Geometric Series

    Does anyone know how to evaluate S_n = \sum_{i=0}^{n-1} i2^i I tried the following. Let r = 2, and figure out the terms in S_n - rS_n Unlike with a regular geometric series, this does not make all but two of the terms disappear. But it does make all but one of the terms turn into a...
  6. S

    Infinite geometric series problem

    Homework Statement Consider the following infinite geometric series: 1 + (2x/3) + (2x/3)^2 + (2x/3)^3 + ... for what values of x does the series converge? Homework Equations i don't know what converge means, i guessed it was for what vlaues does the geometric series is infinite but...
  7. N

    Simplifying a Sum of Squared Terms: A Geometric Series Approach

    Homework Statement How can I simplify sum from j=0 to infinite of x^(2j) ? Homework Equations The Attempt at a Solution THis is close to the geometric series but I'd have to square each individual term
  8. A

    Help me convert Boltzmann distribution/partition function into Geometric series

    Homework Statement 3. The following calculaltion shows how the ratio of e to kT affects the populations of different energy levels. kT is sometimes called the thermal energy; if it is small relative to e, a particle will not be able to access higher energy states. Consider a harmonic...
  9. R

    Quick help in Geometric series question

    Homework Statement The common ratio,ratio,r, of a geometric series is given by: r=\frac{5x}{4+x^2} Find all the values of x for which the series converges Homework Equations The Attempt at a Solution For the series to converge |r|<1 so that |\frac{5x}{4+x^2}|<1 this...
  10. B

    Sum of Complex geometric series

    Homework Statement Use cos ( n * x) = (z ^ n + z ^ -n)/2 to express cos x + cos 3x + cos 5x + ... + cos([2n -1]x) as a geometric series in terms of z. Hence find this sum in terms of x Homework Equations The Attempt at a Solution (z + z^-1)/2 + (z^3 + z^-3)/2 + ... +...
  11. K

    Geometric Series: 2/3^k and -2/10^k

    Homework Statement (infinity)sigma(k = 0) [2(2/6)^k + (-2/10)^k) Homework Equations Geometric Series The Attempt at a Solution I split these up into two geometric series (infinity)sigma(k = 0) [2(1/3)^k] 2 / (1 - 1/3) r = 3 This diverges. (infinity)sigma(k = 0) (-1/5)^k...
  12. P

    Sequences Series geometric series or an arithmetic series?

    This is the sequence: 1, 2, 5, 14, 41, 122 1. Is this a geometric series or an arithmetic series? 2. I know the formula is a sub n=[3^(n-1)+1]/2, but how do you get that from a sub n=a sub 1 * r^(n-1), which is the geometric formula for series.
  13. E

    How Do You Derive the Geometric Series for 1/(1-x)?

    I could use some help with this question: Derive the geometric series representation of 1/(1-x) by finding a0, a1, a2,... such that (1-x)(a0+a1x+a2x^2+a3X^3+...)=1 Thank you.
  14. E

    Infinity geometric series question

    Hi there everyone! Have a quick question for you. The question is: The sum to infinity of a geometric series is 9/2 The second term of the series is -2 Find the value of r, the common ratio of the series. I understand that we have to use the sum to infinity of a geometric series...
  15. K

    Geometric Series: Summing the Powers of x

    Find the sum of the series: \sum\limits_{n = 1}^\infty {nx^n } if \left| x \right| < 1 I thought maybe with the geometric form, but I am not sure.
  16. M

    Is the Sum of a Geometric Series Always Equal to 2?

    Hi Folks, I have this here geometric series which I'm supposed to find the sum of: Given \sum_{n=0} ^{\infty} \frac{2n+1}{2^n} I the sum into sub-sums \sum_{n=0} ^{\infty} 2^{-n} + \sum_{n=0} ^{\infty} \frac{1}{2}^{n-1} taking 2^{-n} Since x^n converges towards 1/1+x therefore I...
  17. J

    What are the possible values of x in this geometric series?

    Hey guys i was having trouble on this question so i was wondering if someone could help me :) In a geometric series, (x-2),(x+5), and (4x-8) are consecutive terms. Determine all possible values of x. :confused:
  18. M

    Summing a Geometric Series: Can We Use the Formula 1/(1-x)?

    Hi Can I claim that in order to find the sum of the series: \sum_{n = 0} ^{\infty} 2^{- n} \sum_{n = 0} ^{\infty} 2^{- n} = \sum_{n = 0} ^{\infty} x^n = \frac{1}{1-x} ? Sincerely Yours Fred
  19. O

    Geometric Series Derivation for Given Identities

    I am trying to derive the geometric series for the following given identities, \begin{array}{l} \frac{1}{{0.99}} = 1.0101010101... \; \; \; {\rm{ (1)}} \\ \frac{1}{{0.98}} = 1.0204081632... \; \; \; {\rm{ (2)}} \\ \end{array} Here is my answer for (1), \sum\limits_{n = 1}^\infty...
  20. M

    Urgend Geometric series question

    Hi I have the following problem: show that 1/(1+x^2)) = 1-x^2 + x^4 + (-1)^n*(x^2n-2) + (-1)^n * (x^2n)/(1+x^2) I that know this arctan function can be expanded as a geometric series by using: 1 + q + q^2 + q^3 + ... + = 1/(1-q) Then by putting q = -x^2. I get...
  21. B

    Verifying Geometric Series Formula: \sum\limits_{k = 0}^N {r^k }

    This has been bothering me for a while. I've seen many different versions of this and I'd just like to get the following cleared up. Is the following true? \sum\limits_{k = 0}^N {r^k } = \frac{{1 - r^{N + 1} }}{{1 - r}} There are other related things I am slightly worried about but I...
  22. U

    How many generations must a person go back to have at least 1000 ancestors?

    Hi everyone, I'm new to these forums, so I've only just realized how much help they can be... I have some questions so please, don't hesitate to aid me in my time of need. These are regarding geometric sequences and series. I'm supposed to be using S=a+ar^n/1-r where s=the sum of the...
  23. Y

    Show a sequence of amounts are a geometric series

    i am given a set of amounts R(1+i)^(n-1)+R(1+i)^(n-2)+... R(1+i)^1,R and so on it has to do with compound interest. how do i prove this is a geometric series?
  24. M

    Proving the Convergence of a Geometric Series with a Tricky Sum Equation

    I've got a problem here... A geometric series has first term 1,the sum of the first 5 terms is twice that of the sum of the 6th to 15th term inclusive. Prove that r^5= \frac{1}{2} \sqrt {3-1} What i did was... 2s_5=s_{15}-s_5 using the formula for the sum of a GS, i got...
  25. P

    Convergence or Divergence of a Geometric Series with r= 1/10

    Evaluate \Sigma 2(1/10)^n or explain why it diverges. (Infinity is on the top of the sum and n=1 on the bottom, I just didn't know how to put it in latex) This was a test question that I got wrong. I thought that it was a geometric series with r= 1/10. This would mean that r is less than 1...
  26. B

    Solving Geometric Series: Find x Values |r|<1

    I am having a little trouble with some questions on geometric series' For example, Find the values of x for which the following geometric series converge I have done the first one easy enough 2+4x+8x^2+16x^3... r=2x |r|<1 |x|<\frac{1}_{2} \frac{-1}{2} < x < \frac{1}_{2} But then it...
  27. A

    Quick HELP GEOMETRIC SERIES SUM

    Help Geometric Sum Help! Plz Hi here is the question It says a retired hockey star wants to set up a scholarship fund to assist an underpriveleged child who would like to go to a post secondary institution. He wants to ensure that the student will have $6000 per year for 5 years. HOw much...
  28. S

    Geom Series: Converge Radius 1, c Complex Num

    how can I write f(z)= 1/(c(1+z)) as a geometric series with radius of convergence 1, where c is a complex number?
  29. K

    How Does the Sum of Alternating Series Lead to a Power Series for 2/(1-x^2)?

    Find a power series for the function centered at c and determine the interval of convergence. c = 0 f(x)=\frac{2}{1-x^2} After some partial fractions work and getting the partials in the form of \frac{a}{1-r} I have \sum x^n + \sum(-x)^n if I factor out the x^n's I get...
  30. P

    Understanding the formula for a geometric series

    I want to understand how the formula for the sum of a geometric sequence is created... This is what I understand so far: A geometric sequence is the sum of a series of numbers, where a term will be multiplied by an amount (the common ratio) to get the next term, and so on... ex...
  31. R

    The expected value of a Geometric Series

    I'm supposed to prove that in a geometric distribution, the expected value, \mu = \frac{1}{p} without the use of moment generating functions (whatever that is) I start off with the very definition of the expected value. \mu_x = E(x) = \sum x \cdot p \cdot (1-p)^{x-1}...
  32. Loren Booda

    How to Solve for the Value of B in a Geometric Series

    Can you solve analytically [oo] [pi] (n)1/n n=1 or [oo] [pi] (n!)1/n! n=1 or [oo] [sum] (1/n)n n=1 or [oo] [sum] (1/n!)n! n=1 ?
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