The convergence Criteria ratio

In summary, the convergence criteria ratio is a measure used in numerical analysis to determine the rate at which a sequence or series approaches a specific value. It is calculated by taking the absolute value of the difference between two consecutive terms and dividing it by the absolute value of the current term. A ratio of 1 indicates that the sequence or series is not converging, while a lower ratio is desirable as it indicates faster convergence. However, there are limitations to using this method, especially if the terms are oscillating or alternating. Other methods may need to be used in these cases.
  • #1
Amaelle
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Homework Statement
is the serie convergent or absolutely convergent?
Relevant Equations
The convergence Criteria ratio
Greetings all

I have a question regarding the convergence criteria ratio, abs(an+1/an) or the n√abs(an) when the limit tend to a value less than 1 does it mean the serie is convergent or absolutely convergent?

Thank you!
 
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  • #2
If the limit is less than 1, the series is absolutely convergent. It is only if the limit equals one that it might be convergent, but not absolutely convergent.
 
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  • #3
FactChecker said:
If the limit is less than 1, the series is absolutely convergent. It is only if the limit equals one that it might be convergent, but not absolutely convergent.
thank you!
 

FAQ: The convergence Criteria ratio

What is the convergence criteria ratio?

The convergence criteria ratio is a measure of how quickly a sequence of numbers approaches its limit. It is used to determine whether a series or sequence is convergent or divergent.

How is the convergence criteria ratio calculated?

The convergence criteria ratio is calculated by taking the absolute value of the quotient of the (n+1)th term and the nth term of a sequence. If this value is less than 1, the sequence is considered convergent.

Why is the convergence criteria ratio important?

The convergence criteria ratio is important because it helps us determine the behavior of a sequence or series. It allows us to identify whether a sequence is approaching a specific value or if it is diverging to infinity.

What is the difference between a convergent and divergent sequence?

A convergent sequence is one that approaches a specific limit, meaning that the terms of the sequence get closer and closer to a single value. A divergent sequence, on the other hand, does not approach a limit and the terms of the sequence either increase or decrease without bound.

How is the convergence criteria ratio used in real-world applications?

The convergence criteria ratio is used in various fields, such as engineering, physics, and economics, to analyze and predict the behavior of systems or processes. It helps in making decisions and solving problems by providing a mathematical framework for understanding the convergence or divergence of a sequence.

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