How to determine convergence/divergence?

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SUMMARY

The equation yt = c1 Lambda1^t + c2 Lambda2^t is analyzed for convergence and divergence. The discussion concludes that the function is convergent only under the condition that both constants θ1 and θ2 fall within the range 0 < θ1, θ2 < 1. This condition ensures that as t approaches infinity, the terms diminish rather than grow, leading to convergence. The exploration of negative values for t was noted but is not the sole determinant of convergence.

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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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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.
 
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:

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