Series Summation: Does the Ratio Test Determine Convergence or Divergence?

  • Thread starter Thread starter barksdalemc
  • Start date Start date
  • Tags Tags
    Series Summation
Click For Summary
SUMMARY

The discussion focuses on the Ratio Test in the context of series summation, specifically addressing the convergence and divergence of sequences Un and Vn. It establishes that if the sequence Vn converges, then the sequence Un also converges, and conversely, if Un diverges, then Vn must also diverge. The participant successfully demonstrated the first part by showing that the ratio Un/Vn is decreasing and that if the limit of this ratio is M, then Un equals MVn. The second part of the problem remains unresolved, indicating a need for further exploration of the ratio properties.

PREREQUISITES
  • Understanding of sequences and series in mathematics
  • Familiarity with the Ratio Test for convergence
  • Knowledge of limits and their properties
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the detailed proof of the Ratio Test for series convergence
  • Explore examples of sequences that illustrate the Ratio Test
  • Investigate alternative convergence tests such as the Root Test
  • Review the implications of the limit comparison test in series analysis
USEFUL FOR

Mathematics students, educators, and anyone studying series convergence and divergence, particularly those interested in advanced calculus or real analysis.

barksdalemc
Messages
54
Reaction score
0
I have this HW problem: Suppose Un and Vn are sequences of positve numbers such that the ratio of Un+1/Un will always we less than Vn+1/Vn. Show that 1) If Vn converges Un converged and 2) If Un diverges, Vn diverges.

I did the first part by showing that for any n, the ration of Un/Vn is decreasing and if lim of the ratio=M then Un=MVn.

Stuck on the second part.
 
Physics news on Phys.org
ration=ratio
 

Similar threads

Interval of convergence of power series from ratio test
  • · Replies 2 ·
Replies
2
Views
2K
Ratio Test vs AST
  • · Replies 4 ·
Replies
4
Views
2K
Proving convergence and divergence of series
  • · Replies 3 ·
Replies
3
Views
2K
On the ratio test for power series
  • · Replies 2 ·
Replies
2
Views
2K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
Series: Determine if they are convergent or divergent
  • · Replies 22 ·
Replies
22
Views
3K
A problem with the convergence of a series
  • · Replies 5 ·
Replies
5
Views
2K
Infinite Series (The Ratio Test)
  • · Replies 3 ·
Replies
3
Views
2K
Intervals of Convergence- Power Series
  • · Replies 11 ·
Replies
11
Views
3K
Does this series converge uniformly?
  • · Replies 7 ·
Replies
7
Views
2K