What is the Limit of the Sequence {[(n+3)/(n+1)]^n} as n Approaches Infinity?

In summary, the problem is finding the limit of the sequence {[(n+3)/(n+1)]^n}, from n=1 to infinity, and the book's answer is e^2. The approach is to use long division to simplify the fraction and get it in the form of the definition of e. However, the legality of dividing the top and bottom by n is questioned. Any help is appreciated.
  • #1
Beamsbox
61
0
Basically,

find the limit of the sequence:

{[(n+3)/(n+1)]^n}, from n=1 to infinity

Book says it's supposed to be e^2, and indeed the graph shows that... I'm not sure what to do with the top of the fraction. Working with the bottom and dividing by n, I obtain, lim as n approaches infinity, (1+(1/n))^n, which is the definition of e... but I'm not sure of the legality of dividing the top and bottom by n, as they're inside the parenthesis to begin with... but if I do it to the top too, I get lim (1+3/n)^n, which I'm not sure what to do with...
lost...

Any help much appreciated!
 
Last edited:
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  • #2
Your title is wrong; this isn't an infinite series, it is a sequence. And although it isn't written that way, I assume it is the whole fraction that is raised to the nth power, not just the denominator.

Hint: Do long division on the fraction on the inside to write it as 1 + (..) and try to get it in the form

(1 + 1/t)t.
 
  • #3
Right, nice assumption, edited and fixed. Long division, I knew I needed it in the form ofthe definition of e, but didn't know how... I'll check it out. Thanks for the help.
 

FAQ: What is the Limit of the Sequence {[(n+3)/(n+1)]^n} as n Approaches Infinity?

1. What are infinite series and why do they have limits?

Infinite series are mathematical expressions that involve an infinite number of terms. They are used to represent a sum of infinitely many numbers. The limit of an infinite series is the value that the series approaches as the number of terms increases towards infinity. This limit helps us understand the behavior of the series as a whole.

2. How do you determine the limit of an infinite series?

There are various methods to determine the limit of an infinite series, such as the Ratio Test, the Root Test, and the Integral Test. These tests involve analyzing the terms of the series and their behavior as the number of terms increases. The limit of the series can then be found using the properties of these tests.

3. Can an infinite series have more than one limit?

No, an infinite series can only have one limit. This is because the limit represents the overall behavior of the series as the number of terms increases towards infinity. If there were more than one limit, the series would not have a well-defined behavior.

4. What are the applications of understanding the limits of infinite series?

The understanding of limits of infinite series has various applications in mathematics and other fields such as physics, engineering, and economics. It is used to approximate functions, solve differential equations, and analyze the convergence or divergence of numerical methods. It also helps in understanding the behavior of systems that involve continuously changing values.

5. Are there any real-life examples of infinite series and their limits?

Yes, there are many real-life examples of infinite series and their limits. One example is the infinite series used to calculate the value of pi, where the limit of the series gives us the exact value of pi. Another example is the infinite series used to model the growth of populations, where the limit represents the maximum possible population size. Infinite series and their limits are also used in finance to calculate compound interest and in signal processing to analyze signals with infinite possibilities.

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