Problem 8 - Calculus: An Intuitive and Physical Approach

Question

Prove rigorously that 1 is not the limit of the sequence for which \(s_{n}=\) \(1 / n\).

Step-by-Step Solution

Answer

No matter how large N is chosen, for any \(n > N\), \(\left| \frac{1}{n} - 1 \right|\) will never be less than every \(\epsilon > 0\), thus 1 cannot be the limit of the sequence \(s_n = \frac{1}{n}\).

Step 1: State the definition of a limit for a sequence

According to the formal definition, a number L is the limit of the sequence \(s_n\) if for every \(\epsilon > 0\) there exists an integer \(N\) such that for all \(n > N\), the absolute value of the difference between \(s_n\) and L is less than \(\epsilon\), mathematically expressed as \(\left| s_n - L \right| < \epsilon\).

Step 2: Apply the definition to prove that 1 is not the limit

Assume that 1 is the limit of the sequence \(s_n = \frac{1}{n}\). According to the definition, this means that for every \(\epsilon > 0\) there must exist an \(N\) such that for all \(n > N\), it holds that \(\left| \frac{1}{n} - 1 \right| < \epsilon\).

Step 3: Show a contradiction

Choose an \(\epsilon\) smaller than 1 (for example, \(\epsilon = \frac{1}{2}\)). For \(n > 1\), \(\left| \frac{1}{n} - 1 \right| = 1 - \frac{1}{n}\), which is always greater than or equal to \(\frac{1}{2}\), contradicting our assumption that \(\left| \frac{1}{n} - 1 \right| < \epsilon\). This shows that no matter how large \(N\) we choose, there will always be an \(n > N\) for which the inequality is not satisfied.

Key Concepts

The Formal Definition of a Limit

Understanding the formal definition of a limit in the context of sequences is essential for students embarking on their journey into calculus and advanced mathematics. The concept of a limit examines what value a sequence approaches as its terms progress towards infinity.

To state it mathematically, a number 'L' is considered the limit of a sequence \(s_n\) if, for every positive number \(\epsilon\), there exists a corresponding integer 'N' such that for all \(n > N\), the terms of the sequence are within an \(\epsilon\)-radius of 'L'. In simpler terms, we can get as close as we want to 'L' by choosing terms beyond a certain point in the sequence.

This condition is represented by the formula \(\left| s_n - L \right| < \epsilon\), and it allows mathematicians to study the behavior of sequences as they extend towards infinity. This definition is crucial for proving many theorems in calculus and serves as a foundation for the continuity of functions, series, and other advanced topics.

Proof by Contradiction

Proof by contradiction is a powerful method of mathematical proof that operates on the principle that a statement can be shown to be true by demonstrating that if the statement were false, it would lead to an absurd or impossible situation.

When we use this technique, we start by assuming the opposite of what we aim to prove. We then logically derive consequences from this assumption. If we arrive at a conclusion that is clearly false or contradictory, then the original assumption cannot be true, and therefore, the statement we aimed to prove must be correct.

In our example case, we assumed that '1' is the limit of the sequence \(s_n = \frac{1}{n}\). By showing that this assumption leads to a direct contradiction with the formal definition of a limit, we have effectively demonstrated that '1' cannot be the limit. Proof by contradiction is not only limited to limits and sequences; it's a fundamental technique used across various mathematical disciplines.

Convergence of Sequences

The concept of convergence is central to understanding sequences and their behavior. A sequence is said to converge if it approaches some limit 'L' as its terms grow without bounds. In essence, no matter how far you go in a converging sequence, the terms get arbitrarily close to the limit.

Convergence is what gives sequences their predictive power and allows us to make meaningful statements about infinite processes. For a sequence to converge, the terms need not reach their limit, but they must become indefinitely close to it as they progress.

For example, the sequence \(s_n = 1/n\) converges to 0. This is because as 'n' grows larger, the value of \(1/n\) gets smaller, edging closer to zero. On the other hand, this sequence obviously cannot converge to 1, as we showed through our proof by contradiction. Through convergence, we have a profound link connecting discrete mathematics with continuous phenomena, a cornerstone in the study of calculus.