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The limit of a sequence is a fundamental concept in calculus and analysis. It describes the value that the terms of a sequence approach as the index goes to infinity. In simpler terms, if you have an infinite sequence of numbers, the limit tells you what single number these terms are getting closer and closer to.

When we refer to the limit of a sequence \( \{a_n\} \), we often denote it as \( \lim_{n \to \infty} a_n \). For a sequence to have a limit, its terms must settle towards a single finite number as \( n \) becomes very large.

In the given problem, the sequence is \( a_n = 1 + \frac{(-1)^n}{n} \). By observing as \( n \to \infty \), the term \( \frac{(-1)^n}{n} \) diminishes to zero. Thus, the limit of the sequence is the constant term, which is 1. In mathematical terms:

• \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} \left( 1 + \frac{(-1)^n}{n} \right) = 1 \).

Understanding limits helps us determine how sequences behave in the long run, which is crucial for analyzing their convergence or divergence.

An alternating sequence is one where the terms switch between positive and negative values. This happens because each term in the sequence changes sign based on whether the index is odd or even.

In our case, this is represented by \( (-1)^n \). For odd values of \( n \), \((-1)^n\) equals -1, and for even \( n \), it equals +1. Consequently, the sequence alternates between adding \( \frac{1}{n} \) and subtracting \( \frac{1}{n} \) from 1.

• At odd \( n \), \( a_n = 1 - \frac{1}{n} \).
• At even \( n \), \( a_n = 1 + \frac{1}{n} \).

Despite these alternating terms, as the sequence progresses towards infinity, the alternating component \( \frac{(-1)^n}{n} \) grows smaller and smaller, driving the entire sequence towards its limit.

Understanding alternating sequences helps identify how oscillations within sequences behave as they head towards certain values or continue without settling on one specific number.

Analyzing whether a sequence converges or diverges is key to understanding its long-term behavior. A sequence converges if its terms get indefinitely close to a specific number, known as the limit. Conversely, a sequence diverges if it fails to approach any definite limit as \( n \to \infty \).

For the sequence in question, \( a_n = 1 + \frac{(-1)^n}{n} \), we observed that as \( n \) increases, the term \( \frac{(-1)^n}{n} \) approaches 0. Hence, regardless of whether \( n \) is odd or even, the sequence draws closer to 1.

Convergence was established through the limit evaluation:

• \( \frac{(-1)^n}{n} \to 0 \)
• Thus, \( 1 + \frac{(-1)^n}{n} \to 1 \)

These conditions confirm convergence because the sequence consistently nears 1, the proposed limit. Divergence would have been suggested if the sequence continued oscillating or growing without commitment to any value.

In summary, convergence and divergence analysis are essential tools for predicting how sequences evolve, highlighting when they stabilize to a limit or spiral without hover.