Assume that bₙ is a sequence of positive numbers converging to 4/5. Determine if the following series converge or diverge.
b. ∑ (from n = 1 to ∞) (5/4)ⁿ (bₙ)

Verified step by step guidance

1. Identify the general term of the series: \(a_n = \left(\frac{5}{4}\right)^n b_n\), where \(b_n\) is a sequence converging to \(\frac{4}{5}\).
2. Since \(b_n \to \frac{4}{5}\) as \(n \to \infty\), analyze the behavior of \(a_n\) by considering the limit \(\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(\frac{5}{4}\right)^n b_n\).
3. Because \(b_n\) approaches a positive constant \(\frac{4}{5}\), the dominant factor in \(a_n\) is \(\left(\frac{5}{4}\right)^n\), which grows exponentially since \(\frac{5}{4} > 1\).
4. Recall that for a series \(\sum a_n\) to converge, the terms \(a_n\) must approach zero as \(n\) approaches infinity. Check if \(\lim_{n \to \infty} a_n = 0\).
5. Since \(\left(\frac{5}{4}\right)^n\) grows without bound and \(b_n\) approaches a positive constant, \(a_n\) does not approach zero. Therefore, by the Divergence Test (also called the nth-term test), the series \(\sum_{n=1}^\infty \left(\frac{5}{4}\right)^n b_n\) diverges.

Key Concepts

Limit of a Sequence

The limit of a sequence is the value that the terms of the sequence approach as n becomes very large. In this problem, knowing that bₙ converges to 4/5 helps us understand the behavior of the terms in the series for large n, which is crucial for analyzing convergence.

Geometric Series and Its Convergence

A geometric series has terms of the form arⁿ, where r is the common ratio. Such a series converges if and only if |r| < 1. Here, the factor (5/4)ⁿ suggests a geometric component with ratio greater than 1, which typically leads to divergence unless modified by other factors.

Comparison Test for Series Convergence

The comparison test involves comparing a given series to a known benchmark series to determine convergence or divergence. Since bₙ approaches 4/5, the terms behave like (5/4)ⁿ times a positive constant, allowing us to compare with a geometric series to conclude about the series' behavior.