Question

Find the limit of the following sequences or state that the limit does not exist. $$\begin{aligned} &\left\\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\right\\}, \text { which is defined by } f(n)=\frac{n-1}{n}, \text { for }\\\ &n=1,2,3, \dots \end{aligned}$$

Short answer

Answer: The limit of the given sequence as n approaches infinity is 1.

Step-by-step solution

Identify the sequence and its limit function

The given sequence is: $$ \left\\{0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\right\\}, $$ Which is defined by the function: $$ f(n) = \frac{n-1}{n} \text{, for } n=1,2,3,\dots $$

Analyze the behavior of the function as n approaches infinity

To find the limit of the sequence, we need to analyze the behavior of f(n) as n approaches infinity. As n grows larger and larger, integer 1 becomes negligibly small as compared to n: $$ \lim_{n\to\infty} f(n) = \lim_{n\to\infty} \frac{n-1}{n} $$

Simplify the function for the limit

To simplify the expression, let's separate the terms: $$ \lim_{n\to\infty} f(n) = \lim_{n\to\infty} \frac{n}{n} - \frac{1}{n} $$ Now we can take the limits of the individual terms: $$ \lim_{n\to\infty} \frac{n}{n} = \lim_{n\to\infty} 1 \\ \lim_{n\to\infty} \frac{1}{n} = 0 $$

Determine the limit of the sequence

Since the limit of the two terms individually are 1 and 0, the limit of the whole function as n approaches infinity is their sum: $$ \lim_{n\to\infty} f(n) = \lim_{n\to\infty} \left( 1 - \frac{1}{n} \right) = 1 - 0 = 1 $$ The limit of the given sequence as n approaches infinity is 1.

Key concepts

Infinite Sequence

When delving into the world of mathematics, an infinite sequence is essentially a list of numbers in a specific order that goes on indefinitely. It’s called 'infinite' because there is no end to the list. A classic example of an infinite sequence is the natural numbers: 1, 2, 3, 4, and so on.

An infinite sequence is often represented by a function like f(n), which provides a formula to find the nth term of the sequence. For instance, if we have f(n) = n/2, this simple formula can give us any term we want in the sequence by simply plugging in the value of n.

Furthermore, understanding sequences is not just about identifying the numbers but also about observing the pattern they follow as n increases. This pattern can be arithmetic (like adding the same number each time) or geometric (where you multiply by the same number), among other possibilities. The sequence in the exercise, \(f(n) = \frac{n-1}{n}\), shows a pattern where the terms gradually get closer to 1 as n increases.

Limit at Infinity

When we talk about the limit at infinity of a sequence, we're trying to describe what happens to the terms of the sequence as they go beyond all bounds — that is, as n gets larger and larger without stopping. It’s like trying to see into the horizon; you’re looking at the trend rather than a specific endpoint because, with infinity, there is no endpoint.

The limit helps us understand the behavior of the sequence in the long run. In mathematical terms, you'll often see it written as \(\lim_{n\to\infty} f(n)\). If this limit exists and is a finite number, it means that the terms of the sequence get closer and closer to that number as n grows. For instance, in the sequence given in the exercise, we observed that as n becomes very large, \(\frac{n-1}{n}\) approaches 1. Hence, \(\lim_{n\to\infty} f(n) = 1\).

If a sequence has a limit at infinity, it adds a layer of predictability to the otherwise endless list of numbers. We may not be able to say what the one millionth term is right away, but we can confidently say what value the terms of the sequence are edging towards.

Convergence of Sequences

Now, let’s crystalize our understanding with the concept of convergence. A sequence is said to converge if it has a limit at infinity that is a real number. That is, if the terms of the sequence get closer and closer to a specific value, then we say the sequence converges to that value.

In our exercise, the sequence \(0, \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \ldots\) converges to 1. This implies a neat property of stability; no matter how far out into the sequence you go, the terms will always be hovering around the number 1. Convergence is a central concept in calculus because it allows us to deal with infinite processes using finite values — a kind of mathematical alchemy that turns the boundless into something manageable.

On the other hand, if a sequence does not get closer to a specific value, we say that it diverges. Understanding whether a sequence converges or diverges, and what it converges to if it does, is incredibly useful in higher mathematics, from summing infinite series to solving differential equations.