Talor series expansion of roots of algebraic equation

Click For Summary
SUMMARY

The discussion focuses on the Taylor series expansion of the roots of the algebraic equation x² - 1 - εx = 0. The roots are expressed as x = ε/2 ± √(1 + ε²/4), and the Taylor series expansion for one of the roots is given as x(1) = 1 + ε/2 + ε²/8 + O(ε³). The author confirms the validity of expanding the root in terms of ε and substituting it back into the algebraic equation, utilizing the binomial theorem to simplify √(1 + ε²/4) into 1 + ε²/8 + O(ε⁴).

PREREQUISITES
  • Understanding of algebraic equations and their roots
  • Familiarity with Taylor series and their applications
  • Knowledge of the binomial theorem
  • Basic calculus concepts related to series expansions
NEXT STEPS
  • Study the derivation of Taylor series for functions
  • Learn about the binomial theorem and its applications in series expansions
  • Explore the implications of perturbation theory in algebraic equations
  • Investigate the convergence of Taylor series in different contexts
USEFUL FOR

Mathematicians, physicists, and engineers interested in algebraic equations, series expansions, and their applications in various fields.

marellasunny
Messages
245
Reaction score
3
I have a algebraic equation like so:
x^2-1-εx=0

the roots are obviously-
x=ε/2±√(1+ε^2/4)

How can I expand the expression for the roots- as a taylor series?

the answer is given as:
x(1)=1+ε/2+ε^2/8+O(ε^3)

I am assuming the author expanded the root 'x' in terms of ε before hand and substituted in the algebraic.Is that even allowed?
 
Physics news on Phys.org
Use the binomial theorem on √(1+ε^2/4) = (1 + ε2/4)1/2 = 1+ε2/8 + O(ε4)
 

Similar threads

Graduate Series expansion of ##(1-cx)^{1/x}##
  • · Replies 3 ·
Replies
3
Views
2K
High School Can We Simplify the Taylor Series Expansion for e^(f(x,y))?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Taylor expansion of f(x)=arctan(x) at infinity
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Keisler Elementary Calculus: computing standard parts of expressions
  • · Replies 26 ·
Replies
26
Views
3K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Method for finding a complex series?
  • · Replies 10 ·
Replies
10
Views
3K
Undergrad Which x_0 to use in a Taylor series expansion?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Classifying Series Summation $$ \sum_{i=0}^{n} 2^{2^i} ~ ?$$
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Infinite Series of Infinite Series
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Does Cauchy's Theorem Confirm That Arithmetic Means of Null Sequences Are Null?
  • · Replies 4 ·
Replies
4
Views
4K