Taylor Series - Range of values

Click For Summary
SUMMARY

The discussion focuses on determining the range of x values for the Taylor series expansion of the exponential function, exp(x). The first four non-zero coefficients of the Taylor expansion are easily identified, but the key issue is understanding the convergence of the series. The ratio test is highlighted as a method to ascertain convergence, indicating that the Taylor series for exp(x) converges for all real numbers. It is established that the Taylor series expansion for exp(x) is valid everywhere.

PREREQUISITES
  • Understanding of Taylor series and their coefficients
  • Familiarity with the exponential function, exp(x)
  • Knowledge of convergence tests, specifically the ratio test
  • Basic calculus concepts, including limits and series
NEXT STEPS
  • Study the application of the ratio test in series convergence
  • Explore the properties of the Taylor series for various functions
  • Investigate the concept of uniform convergence in series
  • Learn about the implications of convergence on function approximation
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus and series, as well as anyone interested in the convergence properties of Taylor series expansions.

wombat4000
Messages
36
Reaction score
0

Homework Statement



im being asked for the first 4 non zero values for the taylor expansion of exp(x) which is simple, but then it asks for the range of x values that are valid for the expansion.

i have never come across ths before - any idea?
 
Physics news on Phys.org
From the first four coefficients of the taylor expansion you can guess what the nth coefficient is going to be. Is there then some test you can use to find out when a sum (don't worry about it being a taylor series!) converges?
 
ratio test? integral test?

but how will working out if the sum convereges help with obtaining the valid x values?
 
The ratio test is useful for calculating when taylor series converge. Each of the terms of your sum is in terms of 'x' so the ratio test tells you when the sum converges in terms of x.
 
the question doesn't mention sum or convergence?
 
The taylor series expansion is going to be valid where the sum converges, or alternatively you can just write that the taylor series expansion for exp is valid everywhere if you've been told this fact.
 

Similar threads

Link between Z-transform and Taylor series expansion
  • · Replies 2 ·
Replies
2
Views
2K
Checking Taylor Series Result of 6x^3-3x^2+4x+5
  • · Replies 13 ·
Replies
13
Views
2K
What is the Taylor expansion of x/sin(ax)?
  • · Replies 5 ·
Replies
5
Views
3K
Find the limit using taylor series
  • · Replies 11 ·
Replies
11
Views
3K
Find Taylor Series from a function and its interval of convergence
  • · Replies 10 ·
Replies
10
Views
2K
How to Evaluate the 8th Derivative of a Taylor Series at x=4
  • · Replies 5 ·
Replies
5
Views
2K
How can the Taylor expansion of x^x at x=1 be simplified to make solving easier?
  • · Replies 2 ·
Replies
2
Views
3K
Performing a Taylor Series Expansion for Lorentz Factor
  • · Replies 13 ·
Replies
13
Views
10K
What Does exp Mean in Mathematical Expressions?
  • · Replies 5 ·
Replies
5
Views
1K
Interval of convergence for Taylor series exp of 1/x^2
  • · Replies 10 ·
Replies
10
Views
4K