D'Alembert Ratio Test: Convergence test

In summary, the D'Alembert Ratio Test is a convergence test that compares the ratio of consecutive terms in a series to a limit, named after the French mathematician Jean le Rond d'Alembert. The test can determine whether a series converges or diverges by checking if the ratio is less than, greater than, or equal to 1. The formula for the D'Alembert ratio involves taking the limit of the absolute value of the quotient of consecutive terms as n approaches infinity. It should be used for series with positive terms that approach zero as n approaches infinity, such as factorial or exponential series. However, the test has limitations, such as not being able to determine the actual sum of a series and
  • #1
ruby_duby
46
0
Hi this is just a general question about using the ratio test for convergence.

If I have to carry out the test to find out if something converges (and I don't need to find out if its absolutely converges, but just convergence), then can my answer to the test be negative?

Or does the formula for the ratio test when finding the limit, use the modulus?
 
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  • #2
Strictly speaking, the ratio test only applies to sequences of positive numbers so, yes, it applies to the modulus (absolute value).
 

Related to D'Alembert Ratio Test: Convergence test

1. What is the D'Alembert Ratio Test?

The D'Alembert Ratio Test is a convergence test used to determine whether an infinite series converges or diverges. It is named after the French mathematician Jean le Rond d'Alembert.

2. How does the D'Alembert Ratio Test work?

The test compares the ratio of consecutive terms in a series to a limit, known as the D'Alembert ratio. If the ratio is less than 1, the series is said to converge. If the ratio is greater than 1, the series diverges. If the ratio is equal to 1, the test is inconclusive and another method must be used.

3. What is the formula for the D'Alembert ratio?

The D'Alembert ratio is calculated as the limit of the absolute value of the quotient of (n+1)th term and nth term of the series, as n approaches infinity. In mathematical notation, it can be written as:
limn→∞ |(n+1)th term / nth term|

4. When should the D'Alembert Ratio Test be used?

The D'Alembert Ratio Test should be used when the series in question has positive terms and the terms of the series approach zero as n approaches infinity. It is most commonly used for series with factorial or exponential terms.

5. What are the limitations of the D'Alembert Ratio Test?

The D'Alembert Ratio Test can only be used to test for convergence or divergence, it cannot determine the actual value of the sum of a series. Additionally, it may give an inconclusive result if the limit of the ratio is equal to 1, in which case another test must be used. It also cannot be used for series with negative terms or terms that do not approach zero as n approaches infinity.

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