55–70. More sequences
Find the limit of the following sequences or determine that the sequence diverges.


{(−1)ⁿ / 2ⁿ}

Verified step by step guidance

1. Identify the general term of the sequence, which is given by \(a_n = \frac{(-1)^n}{2^n}\).
2. Recall that \((-1)^n\) alternates between \(1\) and \(-1\) as \(n\) increases, causing the numerator to alternate signs.
3. Note that the denominator \$2^n\( grows exponentially as \)n$ increases, becoming very large.
4. Consider the behavior of the absolute value of the terms: \(\left|a_n\right| = \frac{1}{2^n}\), which approaches \(0\) as \(n \to \infty\).
5. Since the numerator only changes sign but the magnitude approaches zero, conclude that the sequence converges to \(0\).

Key Concepts

Sequence and Limit

A sequence is an ordered list of numbers defined by a specific formula. The limit of a sequence is the value that the terms approach as the index goes to infinity. Understanding how to find limits helps determine if a sequence converges to a finite value or diverges.

Behavior of Exponential Terms

Exponential terms like 2ⁿ grow rapidly as n increases. When in the denominator, such terms cause the overall fraction to approach zero. Recognizing this behavior is key to evaluating limits involving exponential expressions.

Alternating Sequences

An alternating sequence changes sign with each term, often represented by (−1)ⁿ. While the sign alternates, the magnitude may approach zero or another value. Understanding this helps analyze whether the sequence converges or oscillates without settling.