# Limit of Sequence: Convergent, Limit = 1

• Jolteon
In summary, the sequence ((4^n)/n)^(1/n) is divergent, and the limit is 1. This can be determined by using the property (a/b)^n = (a^n)/(b^n) and the limit lim (n)^(1/n) = 1 as n approaches infinity. However, the inside of the sequence increases faster than the outside exponent decreases, resulting in a divergent sequence.
Jolteon

## Homework Statement

Determine if the following sequence in convergent or divergent, and state the limit if it converges.

((4^n)/n)^(1/n) or "The nth root of '4 to the n' over 'n'"

## Homework Equations

lim ___(n)^(1/n) = 1
n->inf

## The Attempt at a Solution

I had this question on a quiz earlier, but wasn't too sure about my answer. As n approaches infinity, the exponent on the very outside approaches 0. So naturally I thought that anything to the 0 is 1. However, thinking back on it, I think the inside gets larger faster than the outside exponent gets smaller. Thanks for any help, I don't want to wait until Monday!

Try using the property

$$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$

and then the limit you noted.

## 1. What is a limit of a sequence?

A limit of a sequence is the value that the terms of the sequence approach as the index of the terms approaches infinity. It represents the ultimate behavior or end result of the sequence.

## 2. What does it mean for a sequence to be convergent?

A convergent sequence is one in which the terms of the sequence eventually approach and stay close to a specific number (the limit) as the index of the terms increases. In other words, the terms of the sequence get closer and closer to the limit value as the sequence progresses.

## 3. How is the limit of a sequence determined?

The limit of a sequence can be determined by evaluating the behavior of the terms of the sequence as the index approaches infinity. This can be done through various mathematical techniques, such as plotting the terms on a graph or using formulas and rules for specific types of sequences.

## 4. What is the significance of a limit of 1 in a sequence?

A limit of 1 in a sequence means that the terms of the sequence are approaching and staying close to the value of 1 as the sequence progresses. This could indicate that the sequence has a steady or stable behavior and is approaching a specific value or result.

## 5. Can a sequence have a limit of 1 if the terms of the sequence do not actually reach 1?

Yes, a sequence can have a limit of 1 even if the terms of the sequence do not actually reach 1. As long as the terms are getting closer and closer to 1 as the sequence progresses, the limit can be considered 1. This is the concept of approaching but never reaching a limit.

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