Radius of convergence: 1/(1+x^2) about 1, using only real analysis

Click For Summary
SUMMARY

The radius of convergence for the function 1/(1+x^2) expanded about x_0=1 is established as sqrt(2). The discussion emphasizes the necessity of employing real analysis techniques to derive this result. Participants explored various manipulations and geometric series but faced challenges in deriving the Taylor series directly. The suggestion to consider the Taylor series of 1/(1+x) indicates a potential pathway to simplify the problem.

PREREQUISITES
  • Understanding of Taylor series expansion
  • Familiarity with the concept of radius of convergence
  • Knowledge of real analysis principles
  • Experience with geometric series manipulations
NEXT STEPS
  • Study the derivation of Taylor series for 1/(1+x)
  • Research methods for calculating the radius of convergence
  • Explore real analysis techniques for series convergence
  • Investigate geometric series and their applications in convergence tests
USEFUL FOR

Mathematics students, educators, and anyone involved in real analysis or series convergence problems will benefit from this discussion.

pcvt
Messages
17
Reaction score
0
I've seen this thread:
https://www.physicsforums.com/showthread.php?t=297842

and that is the exact question I need to to answer.

What is the radius of convergence of 1/(1+x^2) expanded about x_0=1?

The problem is, I can only use an argument in real analysis.

I see the answer is sqrt(2), but I cannot get that answer.

I've tried a lot of little manipulations to get different geometric series but I can't seem to get it.

Thanks
 
Physics news on Phys.org
What is the taylor series? What tests did you try? It might be easier to find the radius of convergence of 1/(1+x) about 1.
 
I don't see an easy formula for the nth derivative so I'm not sure how to proceed in that direction
 
I didn't do 1/(1+x^2) but there is nice form of the nth derivative of 1/(1+x)
 
Do you mean I can use that to get a taylor series for my function
 

Similar threads

Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
3K
MHB 11.8.4 Find the radius of convergence and interval of convergence
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Question about uniform convergence in a proof
  • · Replies 1 ·
Replies
1
Views
3K
High School How can hyperreal numbers make infinitesimals logically sound in calculus?
  • · Replies 24 ·
Replies
24
Views
6K
MHB Radius of Convergence for $\sum_{j=0}^{\infty} \frac{z^{2j}}{2^j}$
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad What is convergence and 1+2+3+4....... = -1/12
  • · Replies 37 ·
2
Replies
37
Views
4K
MHB What is the Radius of Convergence for zsin(z^2)?
  • · Replies 7 ·
Replies
7
Views
2K
Pointwise convergence for all real x
  • · Replies 5 ·
Replies
5
Views
2K
MHB Taylor Series Expansion and Radius of Convergence for $f(x)=x^4-3x^2+1$
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Why fixed point iteration of ##x^3 = 1-x^2## doesn't converge when ##x_0= 0##
  • · Replies 16 ·
Replies
16
Views
1K