A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. 


{2,3/4,4/9,5/16,…}, which is defined by f(n) = (n+1) / n^2, for n=1,2,3,…

Verified step by step guidance

1. Identify the sequence function: f(n) = \frac{n+1}{n^2}.
2. Consider the limit of the sequence as n approaches infinity: \lim_{n \to \infty} \frac{n+1}{n^2}.
3. Simplify the expression by dividing the numerator and the denominator by n^2: \lim_{n \to \infty} \frac{\frac{n+1}{n^2}}{\frac{n^2}{n^2}} = \lim_{n \to \infty} \frac{\frac{1}{n} + \frac{1}{n^2}}{1}.
4. As n approaches infinity, both \frac{1}{n} and \frac{1}{n^2} approach 0.
5. Conclude that the limit of the sequence is 0.

Key Concepts

Sequences

A sequence is an ordered list of numbers that can be defined by a specific function. Each term in the sequence corresponds to a natural number, and the sequence can be finite or infinite. Understanding sequences is crucial for analyzing their behavior, particularly as the index approaches infinity.

Limits

The limit of a sequence describes the value that the terms of the sequence approach as the index goes to infinity. It is denoted as lim n→∞ f(n) and is fundamental in calculus for determining the behavior of sequences. If the limit exists, it provides insight into the long-term behavior of the sequence.

Limit Laws

Limit laws are rules that govern the evaluation of limits, particularly at infinity. These laws allow for the manipulation of limits in various forms, such as sums, products, and quotients of functions. Applying these laws correctly is essential for finding the limits of sequences and ensuring accurate results.