Calculating Limit: \lambda^t/k with 0<\lambda<1

In summary, calculating the limit in this equation helps determine the value that the expression approaches as t approaches infinity. The value of lambda, between 0 and 1, affects the result of the limit by decreasing the overall value of the expression. It is necessary for k to be a positive number in this equation, as a negative value for k would not be possible when taking the limit as t approaches infinity. This equation can be simplified using logarithms and can be applied in various real-world situations, such as modeling decay or growth processes in science.
  • #1
phonic
28
0
Dear members,

I am calculating the following limit:

[tex]
\lim_{t\rightarrow \infty} \sum_{k=1}^{t-2} \frac{\lambda^{t-k-1}}{k}
[/tex]
where
[tex]
0 < \lambda <1
[/tex]

Does anybody know how to do it? Thanks a lot!
 
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  • #2
A hint:

[tex]0\leq \lim_{t\rightarrow \infty} \sum_{k=1}^{t-2} \frac{\lambda^{t-k-1}}{k}=\lim_{t\rightarrow \infty} \lambda^t \sum_{k=1}^{t-2} \frac{\lambda^{k-1}}{k} \leq \lim_{t\rightarrow \infty} \lambda^t \sum_{k=1}^{\infty} \frac{\lambda^{k-1}}{k}[/tex]
 
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  • #3


Hello,

To calculate this limit, we can use the fact that as t approaches infinity, the sum will approach a Riemann sum for the function f(x) = \frac{\lambda^{x-1}}{x} from k=1 to infinity. This leads to the integral:

\int_{1}^{\infty} \frac{\lambda^{x-1}}{x} dx

Using integration by parts, we can solve this integral to get the final answer of \frac{-\ln\lambda}{1-\lambda}. Therefore, the limit is equal to \frac{-\ln\lambda}{1-\lambda}. I hope this helps! Let me know if you have any further questions.
 

1. What is the purpose of calculating limit in this equation?

The purpose of calculating limit in this equation is to determine the value that the expression approaches as t approaches infinity, which is known as the limit of the expression.

2. How does the value of lambda affect the result of the limit?

The value of lambda, being between 0 and 1, will decrease the overall value of the expression as t approaches infinity. This means that the closer lambda is to 0, the closer the result of the limit will be to 0.

3. Is it necessary for k to be a positive number in this equation?

Yes, it is necessary for k to be a positive number in this equation. This is because a negative value for k would result in a negative value for the expression, which would not be possible when taking the limit as t approaches infinity.

4. Can this equation be simplified further?

Yes, this equation can be simplified using logarithms. By taking the natural logarithm of both sides, the expression becomes ln(\lambda^t/k) = t, which can then be rearranged to solve for t and find the exact value of the limit.

5. How can this equation be applied in real-world situations?

This equation can be applied in various fields of science, such as biology, chemistry, and physics, to model processes that decrease over time. For example, it can be used to calculate the decay of radioactive particles or the growth of bacteria in a controlled environment.

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