Sequences versus series
a.Find the limit of the sequence { (−⁴⁄₅)ᵏ }.

Verified step by step guidance

1. Identify the given sequence as \( a_k = \left(-\frac{4}{5}\right)^k \), where \( k \) is a positive integer.
2. Recall that the limit of a sequence \( a_k \) as \( k \to \infty \) depends on the behavior of the base \( r = -\frac{4}{5} \) raised to the power \( k \).
3. Since \( |r| = \left| -\frac{4}{5} \right| = \frac{4}{5} < 1 \), the terms \( r^k \) approach zero as \( k \to \infty \).
4. Consider the effect of the negative sign: the terms alternate in sign but their magnitude decreases to zero, so the sequence converges to zero.
5. Therefore, the limit of the sequence \( \left\{ \left(-\frac{4}{5}\right)^k \right\} \) as \( k \to \infty \) is zero.

Key Concepts

Sequences and Their Limits

A sequence is an ordered list of numbers defined by a specific formula for its terms. 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 the long-term behavior of sequences.

Geometric Sequences

A geometric sequence is one where each term is found by multiplying the previous term by a constant ratio. The general form is a_k = a * r^k, where r is the common ratio. Recognizing geometric sequences simplifies finding limits and sums.

Limit of a Geometric Sequence with |r| < 1

If the absolute value of the common ratio r in a geometric sequence is less than 1, the terms approach zero as k approaches infinity. This property is crucial for evaluating limits of sequences like (−4/5)^k, where |−4/5| = 0.8 < 1.