45–63. Absolute and conditional convergence Determine whether the following series converge absolutely, converge conditionally, or diverge.
∑ (k = 1 to ∞) (3/4)ᵏ

Verified step by step guidance

1. Identify the given series: \( \sum_{k=1}^{\infty} \left( \frac{3}{4} \right)^k \). This is a geometric series with common ratio \( r = \frac{3}{4} \).
2. Recall that a geometric series \( \sum_{k=1}^{\infty} ar^{k} \) converges if and only if \( |r| < 1 \). Since \( \left| \frac{3}{4} \right| = 0.75 < 1 \), the series converges.
3. Because the terms \( \left( \frac{3}{4} \right)^k \) are positive, the series is absolutely convergent. Absolute convergence means the series of absolute values \( \sum |a_k| \) converges, which is true here since all terms are positive.
4. To confirm, write the sum of the geometric series using the formula for \( |r| < 1 \): \[ S = \frac{a}{1 - r} \], where \( a = \left( \frac{3}{4} \right)^1 = \frac{3}{4} \).
5. Summarize: Since the series converges absolutely, it also converges conditionally by default, but the stronger conclusion is absolute convergence.

Key Concepts

Absolute Convergence

A series ∑a_k converges absolutely if the series of absolute values ∑|a_k| converges. Absolute convergence guarantees convergence regardless of the sign of terms, and it implies the original series converges. This concept helps distinguish between strong convergence and conditional cases.

Geometric Series and Its Convergence

A geometric series has the form ∑r^k, where r is a constant ratio. It converges if and only if |r| < 1, and its sum is given by a/(1-r) for the first term a. Recognizing a series as geometric simplifies determining convergence and sum.

Conditional Convergence

A series converges conditionally if it converges but does not converge absolutely. This means the series ∑a_k converges, but ∑|a_k| diverges. Conditional convergence often occurs in alternating series and requires careful analysis beyond absolute value tests.