Limit of $(x_{n})_{n\geq 1} with Given Conditions

In summary, the conversation discusses finding the limit of a given sequence with variables a, b, and c. The final result is determined to be ac/(c-1)(bc-1), with clarification and adjustments made to the initial setup. The conversation also includes a personal note about initial doubt, which was resolved through further calculation.
  • #1
Vali
48
0
Hi!

I have the following sequence $$(x_{n})_{n\geq 1}, \ x_{n}=ac+(a+ab)c^{2}+...+(a+ab+...+ab^{n})c^{n+1}$$
Also I know that $a,b,c\in \mathbb{R}$ and $|c|<1,\ b\neq 1, \ |bc|<1$
I need to find the limit of $x_{n}$.

My attempt is in the picture.The result should be $\frac{ac}{(1-bc)(1-c)}$
I miss something at these two sums which are geometric progressions.Each sum should start with $1$ but why ? If k starts from 0 results the first terms are $bc$ and $c$ right?
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  • #2
I believe the correct set up is $\displaystyle x_n = \sum_{j=1}^{n+1}\bigg(\sum_{k=0}^{j-1}ab^k\bigg)c^j = \sum_{j=1}^{n+1}\frac{a(b^j-1)}{b-1}c^j$ so that $\displaystyle \lim_{ n \to \infty} x_n = \sum_{j=1}^{\infty}\frac{a(b^j-1)}{b-1}c^j = \frac{ac}{(c-1)(bc-1)}.$
 
  • #3
I'm stupid, I got the correct answer.I just needed to solve some little calculations.I don't know I thought I'm wrong..
Thanks!
 
  • #4
Vali said:
I'm stupid, I got the correct answer.I just needed to solve some little calculations.I don't know I thought I'm wrong..
Thanks!
I see. I didn't go through your calculations because I couldn't zoom in tbh.
 

Related to Limit of $(x_{n})_{n\geq 1} with Given Conditions

1. What is the definition of a limit for a sequence?

A limit for a sequence $(x_{n})_{n\geq 1}$ is a value that the terms of the sequence approach as n gets larger. It can be thought of as the "end behavior" of the sequence.

2. How is the limit of a sequence determined?

The limit of a sequence can be determined by evaluating the behavior of the terms as n gets larger. This can be done by graphing the sequence or using algebraic methods such as finding the pattern in the terms or using the squeeze theorem.

3. What are the conditions for a sequence to have a limit?

A sequence must have a limit if the terms of the sequence approach a single value as n gets larger. Additionally, the sequence must be well-defined, meaning that each term has a unique value and the sequence does not have any "jumps" or discontinuities.

4. Can a sequence have more than one limit?

No, a sequence can only have one limit. If a sequence has multiple limits, it is not well-defined and does not have a true end behavior.

5. How can the limit of a sequence be used in real-world applications?

The limit of a sequence can be used to model real-world situations where a quantity is changing over time. For example, it can be used to predict the future behavior of stock prices or population growth. It is also used in calculus to calculate derivatives and integrals.

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