Part 4/6:
This divergence illustrates an essential concept in sequences: a limit exists only if the terms approach a specific value as ( n ) increases.
Another Alternating Sequence Analysis
Let’s further explore another alternating sequence, ( \frac{(-1)^n}{n} ). Even though the terms oscillate in sign, their absolute values ( \left| \frac{(-1)^n}{n} \right| = \frac{1}{n} ) approach zero. To analyze the behavior mathematically:
[
\lim_{n \to \infty} \frac{1}{n} = 0.
]
Employing Theorem 2, which states that if the limit of the absolute value equals zero, then the limit of the non-absolute value also converges to the same limit, we conclude:
[
\lim_{n \to \infty} \frac{(-1)^n}{n} = 0.
]