27–37. Evaluating series Evaluate the following infinite series or state that the series diverges.
∑ (from k = 0 to ∞)((1/3)ᵏ + (4/3)ᵏ)

Verified step by step guidance

1. Identify the given series as the sum of two separate infinite series: \( \sum_{k=0}^\infty \left( \left(\frac{1}{3}\right)^k + \left(\frac{4}{3}\right)^k \right) = \sum_{k=0}^\infty \left(\frac{1}{3}\right)^k + \sum_{k=0}^\infty \left(\frac{4}{3}\right)^k \).
2. Recognize that each series is a geometric series of the form \( \sum_{k=0}^\infty r^k \), where \( r \) is the common ratio.
3. Recall the convergence criterion for a geometric series: it converges if and only if \( |r| < 1 \). If it converges, the sum is given by \( \frac{1}{1-r} \).
4. Check the common ratios: for the first series, \( r = \frac{1}{3} \), which satisfies \( |r| < 1 \), so it converges; for the second series, \( r = \frac{4}{3} \), which does not satisfy \( |r| < 1 \), so it diverges.
5. Conclude that since the second series diverges, the entire original series diverges and does not have a finite sum.

Key Concepts

Infinite Geometric Series

An infinite geometric series is a sum of terms where each term is obtained by multiplying the previous term by a constant ratio. It converges if the absolute value of the ratio is less than 1, and its sum can be found using the formula S = a / (1 - r), where a is the first term and r is the common ratio.

Convergence and Divergence of Series

A series converges if the sum of its terms approaches a finite limit as the number of terms increases indefinitely. If the terms do not approach zero or the sum grows without bound, the series diverges. Determining convergence is essential before evaluating the sum.

Sum of a Series with Multiple Terms

When a series is composed of sums of multiple sequences, the sum of the series is the sum of the individual series, provided each converges. This allows breaking down complex series into simpler parts that can be evaluated separately and then combined.