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. B

    Infinite Sum of a Geometric Series

    Homework Statement I feel bad asking another question after I just asked one yesterday, but I'm really close this time, I think! I have: \sum_{n=2}^{\infty}\frac{n^2-n}{2^n} And need to find the sum. Homework Equations \sum_{n=1}^{\infty}nx^{n-1}=\frac{1}{(1-x)^2} The Attempt at a...
  2. J

    Convergence and Sum of the Geometric Series: A Quick Guide

    Geometric series problem urgent Homework Statement Calculate the geometric series of Ʃfrom n=1 to infinity of 1/n Homework Equations The Attempt at a Solution I don't know how to start solving, how can I solve this? I have test about this tomorrow I really need some help please.
  3. A

    Exploring Geometric Series Expansion with Higher Powers

    I need to find the solution to the geometric series expansion of the form... \sumn^2*x^n , for n=0,1,2,... most resources I've found only have answers for n*x^n or n*x^(n-1). I have no idea how to calculate this, so I was wondering if there's a book out there that has massive lists of...
  4. O

    Sum to Infinity of a Geometric Series

    Homework Statement Q.: The numbers \frac{1}{t}, \frac{1}{t - 1}, \frac{1}{t + 2} are the first, second and third terms of a geometric sequence. Find (i) the value of t, (ii) the sum to infinity of the series. Homework Equations S\infty = \frac{a}{1 - r} The Attempt at a...
  5. O

    Sum to Infinity of a Geometric Series

    Homework Statement Q.: A geometric series has first term 1 and common ratio \frac{1}{2}sin2\theta. Find the sum of the first 10 terms when \theta = \frac{\pi}{4}, giving your answer in the form h - \frac{1}{2^k}, where h, k \in N. Homework Equations Sn = \frac{a(1 - r^n)}{1 - r}, when...
  6. G

    Calculus II - Infinite Series - Geometric Series

    Homework Statement Hi, I'm trying to solve the problem in the attachment. I was asked to evaluate the left hand side equation of the equal sign. I was unsure how to go about evaluating it so I consulted my solutions manual to look up the first step. The right hand side equation of the...
  7. O

    Sum to Infinity of a Geometric Series

    Homework Statement Q. Find the range of values of x for which the sum to infinity exists for each of these series: (i) 1 + \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + ... (ii) \frac{1}{3} + \frac{2x}{9} + \frac{4x^2}{27} + \frac{8x^3}{81} + ... Homework Equations S\infty =...
  8. O

    Sum to Infinity of a Geometric Series

    Homework Statement Q. Find, in terms of x, the sum to infinity of the series... 1 + (\frac{2x}{x + 1}) + (\frac{2x}{x + 1})^2 + ... Homework Equations S\infty = \frac{a}{1 - r} The Attempt at a Solution S\infty = \frac{a}{1 - r} a = 1 r = U2/ U1 = (\frac{2x}{x + 1})/ 1...
  9. O

    Sum to Infinity of a Geometric Series problem

    Homework Statement Q.: A geometric series has first term a and common ratio r. Its sum to infinity is 12. The sum to infinity of the squares of the terms of this geometric series is 48. Find the values of a and r. Ans.: From textbook: a = 6, r = 1/ 2 Homework Equations...
  10. Femme_physics

    Geometric Series: Find 3 Numbers for 5 Components

    Homework Statement You must enter 3 numbers between 31 and 496 so there will be an increasing geometric series with 5 components. The Attempt at a Solution It tells me I'm off. That q=2. But how? http://img716.imageshack.us/img716/8895/300xk.jpg
  11. C

    Geometric series partial sums question

    I am looking at a geometric series problem that has already been worked out, so not assigned, but I do not see where they get a number: Summation from n=1 to inf: 1/(n^2+4n+3) In doing the partial sums, he has (1/2)* summation... 1/(i+1) - 1/(i+3) I understand the breakup, but where does...
  12. S

    Infinite geometric series application (long)

    Homework Statement Assume that the drug administered intravenously so the concentration of drug in the bloodstream jumps almost immediately to its highest level. The concentration of the drug decays exponentially. A doctor prescribes a 240 milligram (mg), pain-reducing drug to a patient who...
  13. Femme_physics

    Finding the Missing Number in Geometric Series

    Homework Statement http://img833.imageshack.us/img833/681/a1a2.jpg Calculate which number you have to add to a1, a2 and a3 in order to get 3 subsequent numbers in a geometric series The Attempt at a Solution Getting a2 and a3 was easy. Plugging in the values I need for n, I get...
  14. B

    Bit confused about the geometric series

    I'm confused about the sum of the geometric series: \sum ar^{n-1} = \frac{a}{1-r} when |r|<1 but if you have a series like: \sum (1/4)^{n-1} the sum is: \frac{1/4}{1-(1/4)} should't it be \frac{1}{1-(1/4)} because there is no a value?
  15. L

    Convergence of a geometric series; rewriting a series in the form ar^(n-1)

    Homework Statement Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \sumn=1infinity (-3)n-1/4nHomework Equations A geometric series, \sumn=1infinity arn-1=a + ar + ar2 + ... is convergent if |r|< 1 and its sum is \sumn=1infinity arn-1 =...
  16. ThomasMagnus

    Calculating Levels in a Geometric Series Phone Tree

    A school phone tree has 1 person responsible for contacting 3 people. If there are 1500 students in the school, how many levels will there be on the phone tree (assuming 1 person is at the top of the tree)? My Solution: This question forms a geometric series: A(first term)=1 R(common...
  17. A

    Geometric Series Homework: Sum of ((n+1)*3^n)/2^(2n)

    Homework Statement The sum of ((n+1)*3^n)/(2^2n) Homework Equations absolute value of r must be less than 1 for the series to be convergent. The Attempt at a Solution i tried multiplying it out and splitting it up like: 3^n*n/(2^(2n))+3^n/(2^(2n)) but then i am stuck when I...
  18. jegues

    Taylor series using Geometric Series

    Let f(x) = \frac{4-4x}{4x^{2} -8x -5}; given the partial decomposition, \frac{4-4x}{4x^{2} -8x -5} = \frac{1}{5-2x} - \frac{1}{1+2x}, find the Taylor series of f(x) about 1. Express your answer in sigma notation and simplify as much as possible. Dtermine the open interval of...
  19. A

    Simplifying a geometric series with an infinity summation bound

    Homework Statement I am solving some convolutions, and i have come to these solutions. a)\sum2k, summing from -\infty to -1 b)\sum2k, summing from -\infty to n , where n <=-1Homework Equations the geometric series summation formula, from 0 to N \sumak = 1-aN+1 / 1-a , summing from 0 to N The...
  20. M

    Infinite geometric series problem

    Homework Statement \sum_{n=1}^\infty \frac{(-3)^{n-1}}{4^n} The Attempt at a Solution \sum_{n=1}^\infty \frac{(-3)^n-1}{4^n} \frac{1}{4}\sum_{n=1}^\infty \frac(-{3}{4})^{n-1} Can some one please explain how they got from the first step to the 2nd. How do you pull...
  21. jegues

    Taylor Series using Geometric Series and Power Series

    Homework Statement See figure attached. Homework Equations The Attempt at a Solution Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the, \frac{1}{x^{2}} \int x^{-2} = \frac{-1}{x} + C How do I...
  22. jegues

    Exploiting Geometric Series with Power Series for Taylors Series

    I'm confused between some formulae so I'm going to give some examples and you can let me know if what I'm writing is correct. Find the Taylor series for... EXAMPLE 1: f(x) = \frac{1}{1- (x)} around x = 2 Then, \frac{1}{1-(x)} = \frac{1}{3-(x+2)} = \frac{1}{3} \left( \frac{1}{1...
  23. N

    Geometric series. Find the sum of the series. Powers.

    Homework Statement Find the sum of 9 terms of the series 3 + 3^(4/3) + 3^(5/3) + ... Homework Equations I'm just learning sequences and series and senior high school level. I'm finding it hard to apply a, ar, ar^(n-1), ... to this. a = 3. I don't know how to find common...
  24. K

    Find k for Geometric Sequence b1=1000, bn=(2/3)bn-1: 0.001

    b1,b2,b3,... In the geometric sequence above, b1=1000 and bn=(2/3)bn-1 for all n\geq2. What is the least value of k for which bk<0.001? The Attempt at a Solution What I did first was I found what b0 is since we are given b1 and that is 1500. But I do not understand where the k is...
  25. E

    Q4 - Arithmetic and Geometric series

    Homework Statement Let a1, a2, a3 denote the first three terms of a geometrical sequence, for which a1 + a2 + a3 = 26. a1 + 3, a2 + 4, a3 - 3 are the first three terms of an arithmetical sequence. Find the first term and the common quotient (ratio) of the geometrical sequence...
  26. Helios

    Geometric Series: Finding the nth Term

    This looks almost like a geometric series; 1, 2, 5, 14, 41, 122, 365, ... but each term is one less than three times the preceeding one. So is this a sequence or a series? What is a formula for the value of the nth term in terms of n?
  27. Saladsamurai

    Derivative of Geometric Series

    Homework Statement I am having trouble following what is going on in this solution. We are looking to find the expectation value of: f(x,y)=\frac{1}{4^{x+y}}\cdot\frac{9}{16} I have gotten it down to: E(X) = \frac{3}{4}\sum_{x=0}^\infty x\cdot\left(\frac{1}{4}\right)^x\qquad(1) We know...
  28. B

    Evaluating Infinite Geometric Series: a sub n (0.1)^n

    Homework Statement Let an (read 'a sub n') be the nth digit after the decimal point in 2pi+2e. Evaluate SUM (n=1 to inf) an(.1)^n (here, again, an is meant to be 'a sub n') Homework Equations As far as I can see, this is a partial sum of a geometric series. To find the nth...
  29. A

    The Total Vertical Distance of a Ball Dropping from 10 Feet

    a ball is dropped from a height of 10 feet, each bounce is 3/4 of the height of the bounce before a)find an expression for the height hn to which the ball rises after it hits the floor for the nth time so hn= 10(3/4)n b) find an expression for the vertical distance Di the ball has...
  30. M

    Solving Geometric Series: 2*(-1/4)^(n-1)

    {sigma} 2*(-1/4)^(n-1) Could i treat this as a geometric series? i know geometric is in the form of ar^n but the n is (n-1) my A=2 my r= -1/4
  31. Somefantastik

    Summation differentiation geometric series

    Homework Statement find the sum for \sum_{k=1}^{\infty} kx^{k} Homework Equations \sum_{k=0}^{\infty} x^{k} = \frac{1}{1-x}; -1 < x < 1 The Attempt at a Solution \sum_{k=1}^{\infty} kx^{k} = \sum_{n=0}^{\infty}(n+1)x^{n+1} = x\sum_{n=0}^{\infty} (n+1)x^{n} = x \frac{d}{dx}...
  32. Rasalhague

    What is the proof for the geometric series formula?

    \sum_{k=0}^{\infty} ar^k = \frac{a}{1-r} This equation isn't valid, for real numbers, unless \left | r \right | \leq 1. I can see that if r = 1 the denominator is be zero, but what about the other cases? The derivation I've seen is \sum_{k=0}^{\infty} ar^k = \sum_{k=0}^{\infty} ar^k \cdot...
  33. M

    What is the difference between geometric series and laurent series?

    I don't quite understand a few details here. First, What is the difference between geometric series and laurent series? Than, how do I multiply/divide 2 series with each other? Finally, I have this problem, and I'm really clueless as of what to do. Turn 1/(1-cos(z)) into a laurent series.
  34. E

    Sum of Geometric Series: What Am I Doing Wrong?

    I have to find the sum of \sum9(2/3)^n and I get a/1-r where a=9 and r=2/3...but I know a=6 and not 9. Can someone point out to me what I am doing wrong? The sum is from n=1 to infinity. Thanks! EDIT: I am thinking I take a(1) which is 6 as the a in a/(1-r), is this correct?
  35. O

    Rewriting the nth Term of a Geometric Series with Algebra

    What is the algebra required to rewrite the nth term of: (sum from n=0 to infinity) of (pi^n)/(3^n+1) in geometric form?
  36. T

    Infinite sum help (geometric series)

    Homework Statement Well, the original question is to solve this ... \sum 1/(a2 + x2) the sum goes from x=-infinity to infinity (i wasnt sure how to show this with the latex??) and the answer i am supposed to show is \pi/a + (2*\pi/a) * (1/(e2*\pi*a - 1) Homework Equations...
  37. L

    Question on finite and geometric series

    1. 1. Find the exact(no approximations)sum for the finite series S sub n= (2 + 2 + 2(2+...+64 i used the parentheses to represent a radical sign 2. Show that the sum of the first 10 terms of the geometric series 1 + 1/3 + 1/9 + 1/27+... is twice the sum of the first 10 terms of...
  38. R

    Partial sum of geometric series

    ok so i know how to calculate the partial sum of a geometric series. But let's say i only want to calculate the sum of every other term, how would i do this? example: .5^0+.5^1+.5^2+...+.5^n = (.5^(n+1) - 1)/(.5-1) but what equation can i use to get the sum of only these terms...
  39. J

    Geometric Series and Triple Integrals

    Homework Statement \int 1/(1-xyz)dxdydz = \sum1/n3 from n = 1 to infiniti dx 0 to 1 dy 0 to 1 dz 0 to 1 Homework Equations The Attempt at a Solution Not sure how to relate the two of them
  40. S

    Geometric Series - Finding a Partial Sum Equation

    Is it possible to find the partial sum equation for (2^m - 1)/3^m, from m=0 to m=n-1? I know that I'm supposed to rearrange the expression into the format ar^m, so the exponent m must only be on the value r, and not on the constant a. So far the farthest I've gotten is to rearrange it into...
  41. N

    Geometric Series with probability

    Using the formula for the sum of geometric series, show that the values of p(n) sum to 1 p(n)=(1 - \alpha)^n \alpha My attempt: \alpha \sum^\infty_{{\bf n=0}} (1- \alpha)^n I am not sure where to go from here. Any help to show this is true!
  42. F

    Convergence and Sum of Geometric Series - Homework Question

    Homework Statement actually got two questions but both are related so put them in the same place the question asks Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. Inf 1.) E 6(0.9)^(n-1) n=1 Inf (-3)^(n-1)...
  43. P

    Therefore, the simplified function is:f(x) = (1-x)^4

    1. Sum the Geometric Series 1-x+x2-x3+x4 and hence simplify the function [f(x)]4 = 1 - x5 1-x+x2-x3+x4 Homework Equations 3. Not sure I quite get understand this properly, as my attempt doesn't seem quite right. Basically I've gotten...
  44. M

    Finding values of x where the infinite geometric series converge

    Homework Statement (2+x)+(2+x)^2+(2+x)^3 + ... Homework Equations The Attempt at a Solution Ive found that the l r l < 1 the r of this equation is (2 + x) so we have -1 < 2 + x < 1 The values of x where the series coverges is -3 < x < -1 Is this correct...
  45. P

    What Is the Missing Term in the Sequence -1, 5, 2 to Form a Geometric Series?

    If given the values -1, 5, 2 in this sequence, what would be the missing term to make this a geometric series? Also, what would the sum of this geometric series be?
  46. M

    The sum of an infinite geometric series

    Homework Statement 1+(x+1)+(x+1)^2+(x+1)^3 + ... if lx+1l < 1 Homework Equations Sn=a/1-r The Attempt at a Solution My attempt: so I have a = x+1 and r = x+1 from there i get x+1/1-(x+1) which is x+1/1-x-1 from there x+1/-x multiply by the reciprocal my...
  47. R

    Geometric Series Homework: Converge or Diverge? Find Sum

    Homework Statement Does the series from n=1 to infinity of (2)/(n^2-1) converge or diverge? If it converges, find the sum. Homework Equations The Attempt at a Solution I can see right away that the series converges by a limit comparison test by looking at the series. However, to find the sum...
  48. S

    What is the formula for finding the sum of a Geometric Series?

    Hi, I'm having trouble finding the sequence's total sum from a formula concerning Geometric Series. I've been using a calculator to find and manually input all of the terms into a table in Microsoft Excel and adding them all up at the end. The formula that I was given was...
  49. N

    Geometric Series: Questions & Answers

    http://img117.imageshack.us/img117/5258/w1vg1.th.jpg http://img84.imageshack.us/img84/3151/w2px1.th.jpg See above files (one is the question and one is the answer) I can to the whole question, other than the last part - for part (f), why are we concerned with the sum to infinity of...
  50. C

    Math Struggles: Geometric Series & Paying Off a $200 Balance

    What the heck? The minimum monthly payment for a credit card is the larger of $5 or 1/25 of the outstanding balance. If the balance is less than $5, then the entire balance is due. If you make only the minimum payment each month, how long will it take to pay off a balance of $200? Clearly...
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