How to determine convergence/divergence?

[Kinetica]

Homework Statement

Hi.
The equation is: y_t= c₁ Lambda₁^t+c₂ Lambda₂^t

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.

 

[Mark44]

Homework Statement

Hi.
The equation is: y_t= c₁ Lambda₁^t+c₂ Lambda₂^t

This is a little cleaner...
$$y_t = c_1 \lambda_1^t + c_2 \lambda_2^t$$

You can rewrite a^t as (e^{ln (a)})^t = e^t*ln(a), provided that a > 0.

For your equation you need to look at possible restrictions on the two lambdas.

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.

 

[Kinetica]

OK, I found out that the function

c₁e^{t(lnθ1 )} +c₂e^{t(lnθ_2 )} is convergent only when 0<θ1, θ2<1.

All done.
Thanks.