Sequence Behavior for \( a_n = \frac{1}{n^2} \)

We are asked to determine the behavior of the sequence defined by \( a_n = \frac{1}{n^2} \) for \( n > 0 \).

The terms of the sequence are generated by substituting positive integer values for \( n \). Let's look at the first few terms:

• For \( n=1 \), \( a_1 = \frac{1}{1^2} = 1 \)
• For \( n=2 \), \( a_2 = \frac{1}{2^2} = \frac{1}{4} \)
• For \( n=3 \), \( a_3 = \frac{1}{3^2} = \frac{1}{9} \)
• For \( n=4 \), \( a_4 = \frac{1}{4^2} = \frac{1}{16} \)
• ...and so on.

To understand the behavior of the sequence as \( n \) gets larger, we need to examine its limit as \( n \) approaches infinity (\( n \to \infty \)).

Calculating the Limit of the Sequence

A sequence is defined as convergent if its terms approach a specific finite value as \( n \) tends towards infinity. This value is called the limit of the sequence. We calculate this limit for the given sequence:

$ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n^2} $

As \( n \) becomes larger and larger (e.g., 10, 100, 1000, ...), \( n^2 \) grows even faster (100, 10000, 1000000, ...). Consequently, the fraction \( \frac{1}{n^2} \) becomes smaller and smaller, getting arbitrarily close to 0.

Therefore, the limit is:

$ \lim_{n \to \infty} \frac{1}{n^2} = 0 $

Determining Sequence Convergence

Since the limit of the sequence \( a_n = \frac{1}{n^2} \) exists and is equal to a finite number (0), the sequence is classified as convergent.

The sequence is not:

• Divergent: A sequence diverges if its limit is infinite (\( \infty \) or \( -\infty \)) or does not exist.
• Oscillating: A sequence oscillates if its terms do not settle down towards a single value but fluctuate indefinitely.

Because \( \lim_{n \to \infty} \frac{1}{n^2} = 0 \), the sequence clearly converges to 0.