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

Click For Summary

Discussion Overview

The discussion revolves around deriving the geometric series representation of the function 1/(1-x). Participants explore various methods to express this function as a power series, focusing on the coefficients a0, a1, a2, etc., in the context of the equation (1-x)(a0+a1x+a2x^2+a3x^3+...) = 1. The scope includes mathematical reasoning and technical explanation.

Discussion Character

  • Mathematical reasoning
  • Technical explanation
  • Homework-related

Main Points Raised

  • One participant requests assistance in deriving the series and specifies the equation to be solved.
  • Another participant attempts to express the left-hand side of the equation in terms of a power series, suggesting a breakdown of terms.
  • A different participant recommends a method involving multiplication by -x and rearranging terms to approach the solution.
  • One participant suggests using the binomial expansion of 1/(1-x) as an alternative method for deriving the series.
  • Another participant asserts that for |x| < 1, the coefficients a_i can be determined to be 1 for all i, based on the series expansion.
  • A later reply emphasizes the straightforward nature of the task, indicating that it involves direct multiplication and comparison of power series.

Areas of Agreement / Disagreement

Participants present multiple approaches to deriving the series, indicating a lack of consensus on the preferred method. Some methods are more exploratory while others are more direct, reflecting differing perspectives on how to tackle the problem.

Contextual Notes

Participants do not explicitly state assumptions regarding the convergence of the series or the conditions under which the derivations hold true. There is also no resolution on the best approach to take, leaving the discussion open-ended.

Elec68
Messages
2
Reaction score
0
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.
 
Physics news on Phys.org
[tex](1 - x)(a_0 + a_1x + a_2x^2 + a_3x^3 + \dots) = ? = \dots = ? = a_0 + (a_1-a_0)x + (a_2-a_1)x^2 + (a_3-a_2)x^3 + \dots[/tex]
 
I recommend multiplying through by 1, then multiplying through by -x, then rearranging the terms
 
How about (1-x)/(1-x)=1. Expand the denominator into a power series (binomial expansion of 1/(1-x)).
 
If |x| <1, (1-x)*(1+x+x^2+x^3+x^4+...) = 1.
So your coefficients a_i = 1, for all i.
 
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
(1) You (presumably) know how to multiply.
(2) You (presumably) know how to tell when two power series are equal.

There's no "trick" to this one -- you just do exactly what the equation suggests.
 

Similar threads

High School Can You Solve This Exercise on Arithmetic-Geometric Series?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Method for finding a complex series?
  • · Replies 10 ·
Replies
10
Views
3K
MHB Power Series (Which test can i use to determine divergence at the end points)
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Differentiability of a Multivariable function
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad How Does the A/B Relationship Connect to Geometric Series?
  • · Replies 2 ·
Replies
2
Views
1K
MHB How Do You Solve a Geometric Sum with Alternating Signs?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
4K
MHB How do you find the sum of an infinite series?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad How to Derive the Taylor Series for log(x)?
  • · Replies 5 ·
Replies
5
Views
17K