Why does 1/n diverge?

Does 1/n diverge or converge? 

Solution

The sequence \( \frac{1}{n} \) is a classic example in the study of series and sequences within mathematical analysis.

To determine whether the sequence converges or diverges, one must examine the behavior of the sequence as \( n \) approaches infinity.

A sequence converges if its terms approach a specific finite number as \( n \) becomes very large. Conversely, a sequence diverges if its terms do not approach a specific limit as \( n \) increases without bound.

Step-by-Step Analysis of the Sequence \( \frac{1}{n} \):

1. Define the Sequence:

   The sequence in question is \( a_n = \frac{1}{n} \), where \( n \) is a natural number, and \( n \geq 1 \).

2. Examine the Limit:

   To determine convergence or divergence, calculate the limit of \( a_n \) as \( n \) approaches infinity:

   \[ \lim_{{n \to \infty}} \frac{1}{n} \]

3. Apply the Limit:

   As \( n \) increases without bound, the value of \( \frac{1}{n} \) gets closer and closer to 0. This can be formally stated as:

   \[ \lim_{{n \to \infty}} \frac{1}{n} = 0 \]

4. Convergence Criterion:

   Since the terms of the sequence approach 0, a finite number, as \( n \) goes to infinity, the sequence \( \frac{1}{n} \) is said to converge.

5. Conclusion:

   The sequence \( \frac{1}{n} \) converges to 0 as \( n \) approaches infinity.

It is important to note the distinction between the convergence of a sequence and the convergence of a series. While the sequence \( \frac{1}{n} \) converges to 0, the series \( \sum_{n=1}^{\infty} \frac{1}{n} \), also known as the harmonic series, diverges, meaning it does not sum to a finite limit.

This is a fundamental result in the field of series and improper integrals, with significant implications in areas such as harmonic analysis and number theory.