Ratio Test Determines Divergence: 11.6.1

In summary, the problem involves determining the convergence or divergence of a series and the test to be used. W|A suggests using the ratio test, which involves taking the limit of the absolute value of the ratio of consecutive terms. After setting up the first step, the person realizes that using the nth term test would be a simpler approach, as it only requires taking the limit of the terms themselves.
  • #1
karush
Gold Member
MHB
3,269
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Determine Convergence or divergence and test used
$\displaystyle\sum_{n=1}^{\infty} \dfrac{1+4^n}{1+3^n}$
W|A says diverges using ratio test so
$\therefore L=\lim_{n \to \infty}\left|\dfrac{a_n+1}{a_n}\right|>1$
Steps
$\displaystyle L=\lim_{n \to \infty}\left| \dfrac{1+4^{n+1}}{1+3^{n+1}}\cdot\dfrac{1+3^n}{1+4^n}\right|$ ok just seeing if I have this first step set up ok... before I run it thru the grinder.. :cool:
I assume ratio test is a limit test...

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  • #2
I would show divergence using the nth term test, i.e. ...

\(\displaystyle \text{if } \lim_{n \to \infty} a_n \ne 0 \text{ , then }\sum a_n \text{ diverges}\)
 
  • #3
oh... that would save a ton of calculation!
 

What is the ratio test?

The ratio test is a mathematical test used to determine the convergence or divergence of an infinite series. It involves taking the limit of the ratio of consecutive terms of the series and using this limit to determine the behavior of the series.

How is the ratio test used to determine divergence?

The ratio test can be used to determine divergence by checking if the limit of the ratio of consecutive terms is equal to or greater than 1. If this is the case, the series is said to be divergent.

What is the formula for the ratio test?

The formula for the ratio test is:
limn→∞ |an+1/an|
where an represents the nth term of the series.

When is the ratio test inconclusive?

The ratio test is inconclusive when the limit of the ratio of consecutive terms is equal to 1. In this case, the test cannot determine the convergence or divergence of the series and other tests must be used.

What are the limitations of the ratio test?

The ratio test can only be applied to series with positive terms and cannot determine the convergence or divergence of alternating series. It also requires the series to have a specific form, making it unsuitable for more complex series. Additionally, the ratio test may fail to provide a conclusive result for certain series, requiring the use of other tests.

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