How to determine convergence/divergence?

  • Thread starter Kinetica
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In summary, the conversation discusses an equation with constants c1 and c2 and the question of whether it is divergent or convergent. The equation is yt= c1 Lambda1t+c2 Lambda2t, and the attempt at a solution involves rewriting the equation and considering restrictions on the two lambdas. It is found that the function is convergent when 0<θ1, θ2<1.
  • #1
Kinetica
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Homework Statement



Hi.
The equation is: yt= c1 Lambda1t+c2 Lambda2t

where c1 and c2 are constants.
In general, is this equation divergent or convergent?

If divergent, what condition is required to make it convergent?


The Attempt at a Solution


I know that letting t be a negative number, I can make it convergent. But I know that there is more to that answer.

Please help! Thanks.
 
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  • #2
Kinetica said:

Homework Statement



Hi.
The equation is: yt= c1 Lambda1t+c2 Lambda2t
This is a little cleaner...
[tex]y_t = c_1 \lambda_1^t + c_2 \lambda_2^t[/tex]

You can rewrite at as (eln (a))t = et*ln(a), provided that a > 0.

For your equation you need to look at possible restrictions on the two lambdas.
Kinetica said:
where c1 and c2 are constants.
In general, is this equation divergent or convergent?

If divergent, what condition is required to make it convergent?


The Attempt at a Solution


I know that letting t be a negative number, I can make it convergent. But I know that there is more to that answer.

Please help! Thanks.
 
  • #3
OK, I found out that the function

c1et(lnθ1 ) +c2et(lnθ_2 ) is convergent only when 0<θ1, θ2<1.

All done.
Thanks.
 
Last edited:

1. How do you determine if a series converges or diverges?

To determine convergence or divergence of a series, we can use various tests such as the comparison test, ratio test, integral test, or limit comparison test. These tests involve taking the limit of the terms in the series to see if they approach a finite number or infinity. If the limit is a finite number, then the series converges. If the limit is infinity, then the series diverges.

2. What is the difference between absolute and conditional convergence?

Absolute convergence is when the series converges regardless of the order in which the terms are added. On the other hand, conditional convergence is when the series only converges if the terms are added in a specific order. This usually occurs with alternating series.

3. Can a series converge and diverge at the same time?

No, a series cannot converge and diverge at the same time. A series can either converge to a finite value, or diverge to infinity or negative infinity.

4. What is the significance of the terms in a series getting smaller as the series goes on?

In general, as the terms in a series get smaller, the series is more likely to converge. This is because smaller terms indicate that the series is approaching a finite number and is not growing infinitely larger. However, this is not always the case and it is important to use convergence tests to determine the convergence or divergence of a series.

5. Are there any real-world applications of determining convergence/divergence?

Yes, determining convergence and divergence is important in many scientific fields, such as physics, engineering, and economics. In physics, series are used to model physical phenomena and determining if they converge or diverge can provide insight into the behavior of these systems. In economics, series are used to represent economic trends and determining convergence or divergence can help predict future trends. In engineering, series are used to calculate complex systems and determining convergence or divergence can ensure the accuracy of these calculations.

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