How Can the Ratio Test Be Proven for Series Divergence When the Ratio Exceeds 1?

  • Thread starter Thread starter member 731016
  • Start date Start date
  • Tags Tags
    Proof Ratio Test
Click For Summary
The discussion focuses on proving the ratio test for series divergence when the ratio exceeds 1. Participants suggest comparing the series in question to a geometric series as a starting point. An example is provided where if the ratio is greater than 1, one can demonstrate that the series terms will eventually exceed any finite value by considering the addition of a nonzero number infinitely. This approach highlights the fundamental concept that terms growing larger than a converging series will lead to divergence. The conversation emphasizes the need for a clear understanding of series behavior in relation to the ratio test.
member 731016
No Effort - Member warned that some effort must be shown for homework questions.
Homework Statement
Please see below
Relevant Equations
Please see below
For (a) and (b),
1716794684518.png

Does someone please know how to prove this? I don't have any ideas where to start.

Thanks!
 
Physics news on Phys.org
a) b) is standard theory.

Relevant examples also included in the link above.
 
  • Love
Likes member 731016
A good starting point is to compare it to a geometric series. For example if ##c=1/3## can you think of a series that converges whose terms are eventually guaranteed to be larger than the ##x_n##?
 
  • Like
  • Love
Likes member 731016 and FactChecker
If the ratio is > 1, then compare to adding a nonzero number to itself " infinitely often", show it will eventually surpass any finite value.
 
  • Love
Likes member 731016

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
View full post »

Similar threads

Ratio Test vs AST
Replies
4
Views
1K
Interval of convergence of power series from ratio test
Replies
2
Views
1K
On the ratio test for power series
Replies
2
Views
2K
How to show that these sequences are summable?
Replies
1
Views
2K
Proving convergence and divergence of series
Replies
3
Views
1K
Root test proof using Law of Algebra
Replies
6
Views
2K
Which Test to Use: Ratio or Root? Understanding the Convergence of Series
Replies
1
Views
2K
Quick question about Ratio Test for Series Convergence
Replies
2
Views
1K
Series Convergence: What Can the Nth Term Test Tell Us?
Replies
6
Views
2K
Why the series is divergent based on the Preliminary test
Replies
2
Views
1K