How to determine this infinite sum

  • Context: Undergrad 
  • Thread starter Thread starter benf.stokes
  • Start date Start date
  • Tags Tags
    Infinite Sum
Click For Summary
SUMMARY

The discussion focuses on determining the result of the infinite sum represented by the formula (Sum(i=0..+infinity; i*x^i))/(Sum(i=0..+infinity;x^i)) for x<1. The denominator is identified as a geometric series, yielding 1/(1-x). The numerator is derived using elementary calculus, resulting in x/(1-x)^2. Therefore, the final result of the infinite sum is x/(1-x).

PREREQUISITES
  • Understanding of infinite series and convergence
  • Familiarity with geometric series
  • Basic calculus concepts, particularly differentiation
  • Knowledge of limits and their application in series
NEXT STEPS
  • Study the properties of geometric series in detail
  • Learn about differentiation of power series
  • Explore convergence tests for infinite series
  • Investigate applications of infinite sums in calculus
USEFUL FOR

Mathematicians, students studying calculus, and anyone interested in advanced series analysis will benefit from this discussion.

benf.stokes
Messages
67
Reaction score
0
Hi,

How do i determine de result in terms of x of this series:

(Sum(i=0..+infinity; i*x^i))/(Sum(i=0..+infinity;x^i)
For x<1

Thanks
 
Last edited:
Physics news on Phys.org
The denominator is a simple geometric series = 1/(1-x). The numerator series can be gotten using elementary calculus from the denominator series = xd/dx(1/(1-x))=x/(1-x)^2.
Net result = x/(1-x).
 

Similar threads

Undergrad Gabriel's Horn and Sum of Infinite Fractions: Contradiction or Connection?
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Is the following sum a part of any known generalized function?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad The product of 2 infinite sums
  • · Replies 5 ·
Replies
5
Views
2K
High School Arithmetic mean of an infinite number of points?
  • · Replies 20 ·
Replies
20
Views
3K
Undergrad Infinite Series of Infinite Series
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Help required to sum an infinite series in a given equation
  • · Replies 7 ·
Replies
7
Views
2K
Graduate Summing over continuum and uncountable numerocities
  • · Replies 1 ·
Replies
1
Views
3K
MHB How do you find the sum of an infinite series?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Convergence of an infinite series
  • · Replies 15 ·
Replies
15
Views
3K
Undergrad What is the difference between these two power series theorems?
  • · Replies 5 ·
Replies
5
Views
3K