MHB Does the Converse of the Ratio Test Always Hold for Convergent Series?

Click For Summary
The discussion centers on the validity of the converse of the ratio test for series convergence. It is argued that the converse does not hold, with examples provided to illustrate this point. Specifically, the series 0+0+0+... converges, yet its ratio is undefined, suggesting a counterexample. Participants also explore the possibility of non-trivial counterexamples, such as the series defined by a_n = 1/(n(n-1)). The conclusion emphasizes that while a convergent series may have a limit of the ratio less than or equal to 1, the converse of the ratio test is not universally applicable.
alexmahone
Messages
303
Reaction score
0
Is the converse of the ratio test true?
 
Physics news on Phys.org
I don't think so. I think you can construct an easy counterexample. Care to imagine one?
 
Krizalid said:
I don't think so. I think you can construct an easy counterexample. Care to imagine one?

0+0+0+... converges but the ratio is not defined.

I wonder if there are any non-trivial counterexamples.
 
Alexmahone said:
0+0+0+... converges but the ratio is not defined.

I wonder if there are any non-trivial counterexamples.
The "ratio test" says that if $lim \frac{a_{n+1}}{a_n}< 1$ then $\sum a_n$ converges.

The converse is "if $\sum a_n$ converged then $lim \frac{a_{n+1}}{a_n}< 1$".

Find a convergent series such that that limit is 1.
 
Alexmahone said:
0+0+0+... converges but the ratio is not defined.

I wonder if there are any non-trivial counterexamples.
Maybe...

$$ a_n=\frac{1}{n(n-1)} $$
 
HallsofIvy said:
Find a convergent series such that that limit is 1.

$\displaystyle\sum\frac{1}{n^2}$

So, is it safe to say that if a series converges, then $\displaystyle\lim\left|\frac{a_{n+1}}{a_n}\right|\le 1$?
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
View full post »

Similar threads

MHB Series Convergence: Ratio Test & Lim. n→∞
  • · Replies 2 ·
Replies
2
Views
2K
Ratio Test vs AST
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
4K
Interval of convergence of power series from ratio test
  • · Replies 2 ·
Replies
2
Views
2K
How Does the Ratio Test Prove Series Convergence?
  • · Replies 3 ·
Replies
3
Views
3K
MHB Checking Convergence of Series: Inequalities & Tests
  • · Replies 8 ·
Replies
8
Views
2K
MHB Convergence Condition for Applying Ratio Test to Power Series
  • · Replies 2 ·
Replies
2
Views
4K
MHB Proving Convergence of e^z Series Using Ratio Test on Coefficients
  • · Replies 6 ·
Replies
6
Views
5K
MHB Point of convergence of a series
  • · Replies 2 ·
Replies
2
Views
2K
MHB Solving Abs Converge Issue with Ratio Test
  • · Replies 1 ·
Replies
1
Views
2K