Ratio Test for Radius of Convergence | Solving sum(5^n)((x-3)^n)/n"

Click For Summary
SUMMARY

The discussion focuses on applying the Ratio Test to determine the radius of convergence for the series sum(5^n)((x-3)^n)/n. The correct formulation involves the limit expression \lim_{n \to \infty} \left(\frac{5^{n + 1}(x - 3)^{n + 1}}{n + 1}~\frac{n + 1}{5^n (x - 3)^n}\right), which incorporates both the exponential factor and the factorial component. Participants emphasize the importance of including the 5^n factor in the calculations to arrive at the correct conclusion regarding convergence.

PREREQUISITES
  • Understanding of the Ratio Test in calculus
  • Familiarity with series convergence concepts
  • Knowledge of limits and their properties
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the application of the Ratio Test in various series
  • Learn about convergence and divergence criteria for power series
  • Explore the concept of radius of convergence in more depth
  • Practice solving similar problems involving limits and series
USEFUL FOR

Students studying calculus, particularly those focusing on series and convergence, as well as educators looking for examples of the Ratio Test application.

thomas12323
Messages
2
Reaction score
0

Homework Statement



The problem is looking for the radius of convergence
sum(5^n)((x-3)^n)/n
n=1

Homework Equations





The Attempt at a Solution

 
Physics news on Phys.org
What work have you done in using the ratio test?
 
I broke it down to (x-3)n/(n+1)
 
That's not right. What happened to the 5^n factor?
You should have this as part of your work.
\lim_{n \to \infty} \left(\frac{5^{n + 1}(x - 3)^{n + 1}}{n + 1}~\frac{n + 1}{5^n (x - 3)^n}\right )
 

Thread 'Distance between a Clock's hands when the distance is increasing most rapidly'

Question: A clock's minute hand has length 4 and its hour hand has length 3. What is the distance between the tips at the moment when it is increasing most rapidly?(Putnam Exam Question) Answer: Making assumption that both the hands moves at constant angular velocities, the answer is ## \sqrt{7} .## But don't you think this assumption is somewhat doubtful and wrong?
View full post »

Similar threads

Ratio Test vs AST
  • · Replies 4 ·
Replies
4
Views
1K
Interval of convergence of power series from ratio test
  • · Replies 2 ·
Replies
2
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
Does this series converge uniformly?
  • · Replies 7 ·
Replies
7
Views
2K
Understanding the Ratio Test for Series and Its Applications
  • · Replies 7 ·
Replies
7
Views
2K
Ratio Test and Radius of Convergence for ∑ ((n-2)2)/n2, n=1: Homework Solution
  • · Replies 23 ·
Replies
23
Views
2K
Is the Interval of Convergence for (x-2)^n / n^(3n) from -1 to 5?
  • · Replies 2 ·
Replies
2
Views
1K
Series Convergence: What Can the Nth Term Test Tell Us?
  • · Replies 6 ·
Replies
6
Views
2K
Root test proof using Law of Algebra
  • · Replies 6 ·
Replies
6
Views
2K