What is Geometric series: Definition and 182 Discussions

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

    Converging geometric series

    I do not have any reasonable attempts at this problem, as I am trying to figure out how one can get the correct answer when we are not given any values. Maybe if some of you sees a mistake here, that implies that the values from the previous example should be used... ##a_3 = a_1 \cdot k{2}##...
  2. fresh_42

    Changing the Statement Proving $\zeta(2)=\frac{\pi^2}{6}$ via Geometric Series & Substitutions

    Prove $$ \zeta(2) = \sum_{n\in \mathbb{N}}\dfrac{1}{n^2} = \dfrac{\pi^2}{6} $$ by evaluating $$ \int_0^1\int_0^1\dfrac{1}{1-xy}\,dx\,dy $$ twice: via the geometric series and via the substitutions ##u=\dfrac{y+x}{2}\, , \,v=\dfrac{y-x}{2}##.
  3. U

    I Conditional distribution of geometric series

    Can someone help me on this question? I'm finding a very strange probability distribution. Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with...
  4. jisbon

    Proving the Geometric Series with Variable Coefficients: A Scientific Approach

    So this seems to be a geometric Series, but with the coefficients in front, how do I exactly go about proving this? Thanks
  5. U

    Finding the sum of a geometric series

    I'm using the sum of a geometric series formula, but I'm not sure how to find the ratio, r. The n is confusing me. The solution is below, but I'm having trouble with the penultimate step.
  6. Math Amateur

    MHB Convergence of Geometric Series .... Sohrab, Proposition 2.3.8 .... ....

    I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with an aspect of the proof of Proposition 2.3.8 ... Proposition 2.3.8 and its proof read as follows: In the above proof by...
  7. bigbob123

    Calc 2 Sum of Alternating Geometric Series

    A0 = 1 A1 = 3 3(An-1) / 4(An-2) = An
  8. Demystifier

    Insights The Sum of Geometric Series from Probability Theory - Comments

    Greg Bernhardt submitted a new blog post The Sum of Geometric Series from Probability Theory Continue reading the Original Blog Post.
  9. vecsen

    Boas ch1 ex12--the water purification question

    Homework Statement The question is : "In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if n = 2, the water can be made as pure as you...
  10. J

    MHB Real Analysis, Sequences in relation to Geometric Series and their sums

    I will state the problem below. I don't quite understand what I am needing to show. Could someone point me in the right direction? I would greatly appreciate it. Problem: Let p be a natural number greater than 1, and x a real number, 0<x<1. Show that there is a sequence $(a_n)$ of integers...
  11. C

    MHB Geometric Series: Find Sum of Infinity - 9-32-n

    Given that the sum of the first n terms of series, s, is 9-32-n Find the sum of infinity of s. Do I use the formula S\infty = \frac{a}{1-r}?
  12. C

    MHB Solve Geometric Series: Find n from s=9-32-n

    Given that the sum of the first n terms of series, s, is 9-32-n show that the s is a geometric progression. Do I use the formula an = ar n-1? And if so, how do I apply it?
  13. R

    Proving the convergence of series

    Homework Statement Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is: The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞ The question is also attached as a .png file 2. Homework Equations The...
  14. L

    MHB Converging Geometric Series with Negative Values?

    Hiya everyone, Alright ? I have a simple theoretical question. In a decreasing geometric series, is it true to say that the ratio q has to be 0<q<1, assuming that all members of the series are positive ? What if they weren't all positive ? Thank you in advance !
  15. Y

    MHB Geometric Series with Complex Numbers

    Hello all, Three consecutive elements of a geometric series are: m-3i, 8+i, n+17i where n and m are real numbers. I need to find n and m. I have tried using the conjugate in order to find (8+i)/(m-3i) and (n+17i)/(8+i), and was hopeful that at the end I will be able to compare the real and...
  16. F

    MHB Finding sum of infinite geometric series

    find the sum of this infinite geometric series: 1 - √2 + 2 - 2√2 + ... a.) .414 b.) -2.414 c.) series diverges d.) 2 I found that the common difference is 2, so I calculated this: S∞= -.414/-1 s∞= .414 So i got that the answer is A, but will you check this?
  17. C

    Pole of a function, as a geometric series

    Homework Statement Determine the order of the poles for the given function. f(z)=\frac{1}{1+e^z} Homework EquationsThe Attempt at a Solution I know if you set the denominator equal to zero you get z=ln(-1) But if you expand the function as a geometric series , 1-e^{z}+e^{2z}... I...
  18. E

    Values of x for which a geometric series converges

    Need help with a homework question! The question gives: The first three terms of a geometric sequence are sin(x), sin(2x) and 4sin(x)cos^2(x) for -π/2 < x < π/2. First I had to find the common ratio which is 2cos(x) Then the question asks to find the values of x for which the geometric series...
  19. binbagsss

    Geometric series algebra / exponential/ 2 summations

    Homework Statement I want to show that ## \sum\limits_{n=1}^{\infty} log (1-q^n) = -\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty} \frac{q^{n.m}}{m} ##, where ##q^{n}=e^{2\pi i n t} ## , ##t## [1] a complex number in the upper plane.Homework Equations Only that ## e^{x} =...
  20. karush

    MHB Use the techniques of geometric series

    $\tiny{242.WS10.a}$ \begin{align*} &\textsf{use the techniques of geometric series} \\ &-\textsf {telescoping series, p-series, n-th term } \\ &-\textsf{divergence test, integral test, comparison test,} \\ &-\textsf{limit comparison test,ratio test, root test, } \\ &-\textsf {absolute...
  21. karush

    MHB 10.3.54 repeating decimal + geometric series

    $\tiny{206.10.3.54}$ $\text{Write the repeating decimal first as a geometric series} \\$ $\text{and then as fraction (a ratio of two intergers)} \\$ $\text{Write the repeating decimal as a geometric series} $ $6.94\overline{32}=6.94323232 \\$ $\displaystyle A.\ \ \...
  22. karush

    MHB Series using Geometric series argument

    $\displaystyle\text{if} \left| r \right|< 1 \text{ the geometric series } a+ar+ar^2+\cdots ar^{n-1}+\cdots \text{converges} $ $\displaystyle\text{to} \frac{a}{(1-r)}.$ $$\sum_{n=1}^{\infty}ar^{n-1}=\frac{a}{(1-r)}, \ \ \left| r \right|< 1$$ $\text{if} \left| r \right|\ge 1 \text{, the series...
  23. Pouyan

    Differential equations and geometric series

    Homework Statement I Have a differential equation y'' -xy'-y=0 and I must solve it by means of a power series and find the general term. I actually solved the most of it but I have problem to decide it in term of a ∑ notation! Homework Equations y'' -xy'-y=0 The Attempt at a Solution I know...
  24. karush

    MHB Calculate the sum for the infinite geometric series

    Calculate the sum for the infinite geometric series $4+2+1+\frac{1}{2}+...$ all I know is the ratio is $\frac{1}{2}$ $\displaystyle\sum_{n}^{\infty}a{r}^{n}$ assume this is used
  25. alexandria

    Arithmetic and Geometric Series

    Homework Statement Homework Equations no equations required 3. The Attempt at a Solution a) so for part c) i came up with two formula's for the tortoise series: the first formula (for the toroise series) is Sn = 20n This formula makes sense and agrees with part a). for example, if the...
  26. Biker

    B Arithmetic Series and Geometric Series

    Here is a question that I have a problem with, It doesn't seem to have a solution: An increasing sequence that is made of 4 positive numbers, The first three of it are arithmetic series. and the last three are geometric series. The last number minus the first number is equal to 30. Find the sum...
  27. R

    Heat conduction problem in a ring of radius a

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  28. Drakkith

    I Geometric Series Convergence and Divergence

    I'm a little confused on geometric series. My book says that a geometric series is a series of the type: n=1 to ∞, ∑arn-1 If r<1 the series converges to a/(1-r), otherwise the series diverges. So let's say we have a series: n=1 to ∞, ∑An, with An = 1/2n An can be re-written as (1/2)n, which...
  29. The-Mad-Lisper

    Preliminary Test of Alternating Geometric Series

    Homework Statement \lim_{n \to \infty}\frac{(-1)^{n+1} \cdot n^2}{n^2+1} Homework Equations \lim_{n \to \infty}a_n \neq 0 \rightarrow S \ is \ divergent The Attempt at a Solution I tried L'Hopital's rule, but I could not figure out how to find the limit of that pesky (-1)^{n+1}. Edit: This...
  30. NihalRi

    Use geometric series to write power series representation

    Homework Statement Complete the proof that ln (1+x) equals its Maclaurin series for -1< x ≤ 1 in the following steps. Use the geometric series to write down the powe series representation for 1/ (1+x) , |x| < 1 This is the part (b) of the question where in part (a)I proved that ln (1+x)...
  31. T

    Finite geometric series formula derivation? why r*S?

    what is the rationale of multiplying "r" to the second line of series? why does cancelling those terms give us a VALID, sound, logical answer? please help. here's a video of the procedure
  32. Matejxx1

    Find the limit of a sequence

    Homework Statement given a geometric sequence sin(x),sin(2x), . . . c) find for which values of x∈(0,π) this sequence converges and calculate its limit Homework Equations |q|<1 or -1<q<1The Attempt at a Solution Ok so in part a) and b) i calculated the quotient and found out that...
  33. P

    Infinite geometric series

    Homework Statement hello this question is discussed in 2009 but it is closed now If you invest £1000 on the first day of each year, and interest is paid at 5% on your balance at the end of each year, how much money do you have after 25 years? Homework Equations ## S_N=\sum_{n=0}^{N-1} Ar^n##...
  34. P

    Infinite series Geometric series

    Homework Statement hello i have a question to this solved problem in the book " Mathematical Methods for Physics and Engineering Third Edition K. F. RILEY, M. P. HOBSON and S.J. BENCE " page 118 Consider a ball that drops from a height of 27 m and on each bounce retains only a third of its...
  35. T

    Geometric series and its derivatives

    Homework Statement I was browsing online and stumbled upon someone's explanation as to why 1 -2 +3 -4 + 5... towards infinity= 1/4. His explanation didn't make sense to me. He starts with a geometric series, takes a derivative, and plugs in for x = -1, and gets a finite value of 1 -2 + 3 - 4...
  36. M

    MHB Convergence of a geometric series

    Hi everyone, I am generally familiar with convergent series. However, in one economics paper (Becker&Tomes 1979), I found the following that confuses me:$$\sum_{j=0}^{k} \beta^{j} h^{k-j} = \beta^{k}(k+1)\quad \text{if} \quad\beta =h$$ however, $$\sum_{j=0}^{k} \beta^{j} j^{k-j} =...
  37. N

    The webpage title could be: Solving for x in an Infinite Geometric Series

    x+x^2+x^3+x^4... = 14 Find x Could someone please provide an explanation. Thank you
  38. N

    Infinite geometric series

    x+x^2+x^3+x^4... = 14 Find x Could someone please provide an explanation on how to solve this?
  39. MidgetDwarf

    Cannot follow this argument pertaining to geometric series.

    In order to construct a geometric series we do the following: chose a number a such that a does not equal 0 and a second number r that is between (-1,1). We call r the ration because it is the ratio, the progression of each term to its predecessor. We have An=a+ar+ar^2+ar^3...ar^n We multiply...
  40. mpx86

    Probably a geometric series question

    Homework Statement 2^32 – (2 + 1) (2^2 – 1) (2^4+1) (2^8+1) (2^16+1)} is equal to Homework EquationsThe Attempt at a Solution Solved it by opening the bracket Answer: 2^31 + 2^24 + 2^ 18 - 2^7 + 2 Option' are 0 1 2 2^16 None of the options matched... Is there a mistake in question statement...
  41. P

    Taylor series with using geometric series

    The question is: Determine the Taylor series of f(x) at x=c(≠B) using geometric series f(x)=A/(x-B)4 My attempt to the solution is: 4√f(x) = 4√A/((x-c)-B = (4√A/B) * 1/(((x-c)/B)-1) = (4√A/-B) * 1/(1-((x-c)/B)) using geometric series : 4√f(x) = (4√A/-B) Σ((x-c)/B)n f(x)= A/B4 *...
  42. Destroxia

    How does a geometric series converge, or have a sum?

    Homework Statement How does a geometric series have a sum, or converge? Homework Equations Sum of Geometric Series = ##\frac {a} {1-r}## If r ≥ ±1, the series diverges. If -1 < r < 1, the series converges. The Attempt at a Solution How exactly does a infinite geometric series have a sum...
  43. W

    Sum of a geometric series of complex numbers

    Homework Statement Given an integer n and an angle θ let Sn(θ) = ∑(eikθ) from k=-n to k=n And show that this sum = sinα / sinβ Homework Equations Sum from 0 to n of xk is (xk+1-1)/(x-1) The Attempt at a Solution The series can be rewritten by taking out a factor of e-iθ as e-iθ∑(eiθ)k from...
  44. G

    Finding the height of a ball with a geometric series

    Homework Statement A ball is dropped from one yard and come backs up ##\dfrac{2}{3}## of the way up and then back down. It comes back and ##\dfrac{4}{9}## of the way. It continues this such that the sum of the vertical distance traveled by the ball is is given by the series...
  45. M

    Sum of Geometric Series by Differentiation

    Homework Statement Find the sum of the following series: Σ n*(1/2)^n (from n = 1 to n = inf). Homework Equations I know that Σ r^n (from n = 0 to n = inf) = 1 / (1 - r) if |r| < 1. The Attempt at a Solution [/B] I began by rescaling the sum, i.e. Σ (n+1)*(1/2)^(n+1) (from n = 0 to n =...
  46. M

    Infinite series of sin + cos both to the 2n power

    Homework Statement For the following series ∑∞an determine if they are convergent or divergent. If convergent find the sum. (ii) ∑∞n=0 cos(θ)2n+sin(θ)2n[/B]Homework Equations geometric series, [/B]The Attempt at a Solution First I have to show that the equation is convergent. Both cos(θ)...
  47. R

    Finding Value of Sum of Geometric Series

    Homework Statement Let ## S_k , k = 1,2,3,…,100 ## denote the sum of the infinite geometric series whose first term is ## \frac{k-1}{k!} ## and the common ratio is ##\frac {1}{k}##. Then value of ##\frac {100^2}{100!} + \sum\limits_{k=1}^{100} | (k^2 - 3k + 1)S_k | ## is Homework Equations...
  48. RJLiberator

    Geometric Series Sum Question

    Homework Statement I am giving the sum: k=1 to infinity Σ(n(-1)^n)/(2^(n+1)Homework Equations first term/(1-r) = sum for a geometric series The Attempt at a Solution [/B] With some manipulation of the denominator 2^(n+1) = 2*2^n I get the common ratio to be (-1/2)^n while the coefficient is...
  49. I

    Geometric series vs. future value computation based on geometric series

    Homework Statement Hello! Revising geometric series, I have understood that I have the following issue - I have read again about these series and, please, take a look at what I have gotten as a result (picture attached). If I calculate a future cash flow, that is I take, for example, a = 0,5...
  50. T

    Understanding an expansion into a geometric series

    Hi all, I am reading through Riley, Hobson, and Bence's Mathematical Methods for Phyisics and Engineering, and on page 854 of my edition they describe (I am replacing variables for ease of typing) "expanding 1/(a-z) in (z-z0)/(a-z0) as a geometric series 1/(a-z0)*Sum[((z-z0)/(a-z0))^n] for n...
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