Proving the Limit of a Sequence Approaches Zero: Techniques and Applications

Understanding the limit of a sequence as n approaches infinity is a fundamental concept in mathematical analysis. In particular, demonstrating that (a_n) approaches 0 as (n) goes to infinity is a common and important problem. There are several methods to tackle this, each with its unique advantages and applications. Let's explore these techniques in detail.

1. Direct Comparison

Description: This method involves comparing the terms of the sequence (a_n) with a sequence of positive numbers that converge to 0, such as (frac{1}{n}) or (frac{1}{n^2}). If (a_n leq frac{c}{n}) or (a_n leq frac{c}{n^2}) for all (n geq N) and some constant (c > 0), then we can conclude that (lim_{n to infty} a_n 0).

Example: Consider the sequence (a_n frac{2}{n^2}). Here, (a_n leq frac{1}{n^2}) for all (n geq 1). Since (lim_{n to infty} frac{1}{n^2} 0), by the direct comparison test, we have (lim_{n to infty} a_n 0).

2. Squeeze Theorem (Sandwich Theorem)

Description: The Squeeze Theorem is a powerful tool in dealing with limits. If we can find two sequences (b_n) and (c_n) such that (b_n leq a_n leq c_n) for all (n) and both (b_n) and (c_n) converge to 0 as (n to infty), then (lim_{n to infty} a_n 0).

Example: Consider the sequence (a_n sinleft(frac{1}{n}right)). We know that (-1 leq sin(x) leq 1) for all (x). Thus, for (n geq 1), (-1 leq sinleft(frac{1}{n}right) leq 1). However, as (n to infty), both (b_n -1) and (c_n 1) approach 0. Therefore, by the Squeeze Theorem, (lim_{n to infty} sinleft(frac{1}{n}right) 0).

3. Algebraic Manipulation

Description: This method involves rewriting the sequence (a_n) in a form that makes it easier to see that the limit is 0. This often involves simplifying expressions or applying limits to individual components.

Example: Consider the sequence (a_n 1 - frac{1}{n}). We can directly see that as (n to infty), (frac{1}{n} to 0). Therefore, (lim_{n to infty} (1 - frac{1}{n}) 1 - lim_{n to infty} (frac{1}{n}) 1 - 0 1). However, if the goal was to show that a sequence approaches 0, we might manipulate it differently. For example, consider a sequence like (a_n frac{1}{1 n^2}). As (n to infty), the denominator grows without bound, making the entire expression approach 0.

4. Limit Definition

Description: The formal definition of the limit involves proving that for every (epsilon > 0), there exists an (N) such that (|a_n - 0| for all (n geq N). This defines the limit of the sequence as 0.

Example: For the sequence (a_n frac{1}{n}), we need to show that for any (epsilon > 0), there exists an (N) such that (|frac{1}{n} - 0| for all (n geq N). This simplifies to (frac{1}{n} , which can be achieved by choosing (N > frac{1}{epsilon}). Therefore, by the formal definition of limit, we have (lim_{n to infty} frac{1}{n} 0).

Conclusion

Proving that a sequence (a_n) approaches 0 as (n) approaches infinity can involve a variety of methods, including direct comparison, the Squeeze Theorem, algebraic manipulation, and the formal limit definition. Choosing the appropriate method depends on the specific form of the sequence. A combination of these techniques often proves to be the most effective approach.