Convergence of Series: Find the Value | Online Databases

  • Context: Undergrad 
  • Thread starter Thread starter Eismc[]
  • Start date Start date
  • Tags Tags
    Convergence Series
Click For Summary
SUMMARY

The value of the series SUM (k from 0 to infinity) k²(x^k) for |x| < 1 can be derived using calculus techniques. Starting with the geometric series sum x^k = 1/(1-x), one can differentiate both sides, multiply by x, and repeat the process to arrive at the desired result. This method is more efficient than searching through online databases for the answer.

PREREQUISITES
  • Understanding of infinite series and convergence
  • Familiarity with calculus, specifically differentiation
  • Knowledge of geometric series and their properties
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the derivation of the geometric series and its applications
  • Learn advanced techniques in calculus, focusing on series differentiation
  • Explore online databases for mathematical series, such as Wolfram Alpha
  • Investigate convergence tests for infinite series
USEFUL FOR

Mathematicians, students studying calculus, educators teaching series convergence, and anyone interested in advanced mathematical concepts.

Eismc[]
Messages
4
Reaction score
0
Hello everyone,

does someone know what the value of [SUM (k from 0 to infinity) k²(x^k)] (with |x| < 1) is? I couldn't find it anywhere. :-( Are there any good online databases for such things?

Many thanks in advance!
 
Physics news on Phys.org
Starting with sum xk =1/(1-x), you can easily derive what you want faster than trying to look it up. Take the derivative of both sides, mutiply both sides by x, repeat both steps, you will end up with what you are looking for.
 

Similar threads

Undergrad Infinite Series - Divergence of 1/n question
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Infinite Series of Infinite Series
  • · Replies 7 ·
Replies
7
Views
3K
High School Difficulty with understanding whether 1/n converges or diverges
  • · Replies 11 ·
Replies
11
Views
6K
MHB Power Series (Which test can i use to determine divergence at the end points)
  • · Replies 1 ·
Replies
1
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
Undergrad Uniform convergence of family of functions with continuous index
  • · Replies 3 ·
Replies
3
Views
3K
Graduate How Can We Ensure Convergence in Function Approximations Beyond Taylor Series?
  • · Replies 23 ·
Replies
23
Views
4K
Undergrad Taming a Divergent Series -- But how does it work?
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Looking for references on this form of a Taylor series
  • · Replies 3 ·
Replies
3
Views
2K