# Percise defination of the limit of a sequence problem

• GNelson
N = (1/10)^2 = 1/100.In summary, the problem is to find a natural number N such that for any n greater than N, the absolute value of ln(n)/n - 0 is less than 1/10. One way to solve this is by choosing an upper bound for ln(n), such as sqrt(n), and then setting N equal to the square of the reciprocal of the desired accuracy, in this case 1/10.
GNelson

## Homework Statement

it is shown that lim n->Infinity of ln(n)/n=0
Find a natural number N such that
n> N -> |ln(n)/n - 0| < 1/10

## The Attempt at a Solution

A sequence has a limit 0 if for every $$\epsilon$$>0 there exists a number N such that for every n > N

|ln(n)/n-0|<$$\epsilon$$

Take $$\epsilon$$=1/10, as ln(n)/n > 0 for any sufficently large n we have

ln(n)/n < 1/10.

so I choose N=10ln(n).

My problem starts here this is a function of a variable being sent to infinity I am not sure how exactly one solves for N that is purely a function of $$\epsilon$$.

You can find any natural number, so just get an upper bound for ln(n).

For example, ln(n) < sqrt(n)

## 1. What is the definition of the limit of a sequence problem?

The limit of a sequence problem is a concept in mathematics that refers to the behavior of a sequence of numbers as the terms in the sequence approach a specific value, known as the limit. It is used to determine the behavior and convergence of a sequence and is a fundamental concept in calculus and analysis.

## 2. How is the limit of a sequence problem calculated?

The limit of a sequence problem is calculated by taking the limit of the terms in the sequence as the number of terms approaches infinity. This is typically done by evaluating the terms of the sequence and looking for patterns or using mathematical techniques such as the squeeze theorem or the ratio test.

## 3. What is the difference between a sequence and a series?

A sequence is a list of numbers in a specific order, while a series is the sum of a sequence of numbers. In other words, a sequence is a list of terms, while a series is the sum of those terms. The limit of a sequence problem focuses on the behavior of the individual terms in a sequence, while the limit of a series problem focuses on the overall sum of those terms.

## 4. What is the importance of understanding the limit of a sequence problem?

The limit of a sequence problem is an important concept in mathematics because it allows us to determine the behavior of a sequence as the number of terms increases. It is used in many real-world applications, such as analyzing the growth of populations, the convergence of algorithms, and the behavior of financial investments.

## 5. Can you give an example of a sequence with a limit?

One example of a sequence with a limit is the sequence of reciprocals: 1, 1/2, 1/3, 1/4, ... As the number of terms increases, the terms get smaller and approach 0, making the limit of the sequence 0. In other words, as n approaches infinity, the limit of the sequence 1/n is 0.

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