Help me find the radius of convergence?

Click For Summary
SUMMARY

The radius of convergence for the series Ʃn!(x-1)n, evaluated using the ratio test, is determined to be 0. The calculation shows that as n approaches infinity, the limit L = lim(n→∞)(n+1) results in infinity, leading to the conclusion that the series converges only at x = 1. This indicates that the series diverges for all other values of x.

PREREQUISITES
  • Understanding of series and convergence concepts
  • Familiarity with the ratio test for convergence
  • Basic knowledge of factorial notation and its properties
  • Experience with limits in calculus
NEXT STEPS
  • Study the ratio test in more detail to understand its applications
  • Learn about different convergence tests for series, such as the root test
  • Explore the concept of power series and their convergence properties
  • Review factorial growth rates and their impact on series convergence
USEFUL FOR

Students studying calculus, particularly those focusing on series and convergence, as well as educators looking for examples of convergence tests in action.

madcattle
Messages
10
Reaction score
0

Homework Statement


Ʃn!(x-1)n
I need to find the radius of convergence for this summation from n=0 to n=∞


The Attempt at a Solution


I started off with the ratio test:

(n!(n+1)(x-1)(x-1)n)/(n!(x-1)n) = (n+1)(x-1)

(x-1)lim(n+1)...Now at this point it looks to me like the series does not actually converge, but my book is telling me that it does. Am I looking at something the wrong way? Not actually understanding the problem?
 
Physics news on Phys.org
It looks like it converges only at x = 1.
 
Last edited:
Usually you do the ratio test on the part of the sum that doesn't have the factor with x, so in this case with just n!. Simplifying that gets you L = limn→∞(n+1) = ∞ and the radius of convergence is r = 1/L = 0.

The only x value where it converges is x=1.
 

Similar threads

Interval of convergence of power series from ratio test
  • · Replies 2 ·
Replies
2
Views
2K
Pointwise convergence for all real x
  • · Replies 5 ·
Replies
5
Views
2K
Does this series converge uniformly?
  • · Replies 7 ·
Replies
7
Views
2K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
On the ratio test for power series
  • · Replies 2 ·
Replies
2
Views
2K
Ratio Test vs AST
  • · Replies 4 ·
Replies
4
Views
1K
The convergence Criteria ratio
  • · Replies 2 ·
Replies
2
Views
1K
Proving convergence of rational sequence
  • · Replies 2 ·
Replies
2
Views
2K
Addition of power series and radius of convergence
  • · Replies 14 ·
Replies
14
Views
2K
Root test proof using Law of Algebra
  • · Replies 6 ·
Replies
6
Views
2K