Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The sum ∑ (k = 1 to ∞) 1 / 3ᵏ is a p-series.

Verified step by step guidance

1. Recall the definition of a p-series: a series of the form \(\sum_{k=1}^{\infty} \frac{1}{k^p}\), where \(p\) is a positive real number.
2. Examine the given series: \(\sum_{k=1}^{\infty} \frac{1}{3^k}\). Notice that the denominator is \$3^k\(, which is an exponential expression, not a power of \)k$.
3. Since the denominator is not of the form \(k^p\), the series does not fit the definition of a p-series.
4. Instead, the given series is a geometric series with common ratio \(r = \frac{1}{3}\), because each term is obtained by multiplying the previous term by \(\frac{1}{3}\).
5. Therefore, the statement that the series is a p-series is false; it is actually a geometric series.

Key Concepts

Definition of a p-series

A p-series is an infinite series of the form ∑ 1/n^p, where n starts at 1 and p is a positive constant. It converges if and only if p > 1. Recognizing whether a series fits this form is essential to classify it as a p-series.

Geometric series

A geometric series is an infinite sum where each term is a constant ratio r times the previous term, expressed as ∑ ar^k. It converges if |r| < 1, and its sum can be calculated using the formula a/(1-r). Identifying geometric series helps distinguish them from p-series.

Series classification and counterexamples

Determining whether a series belongs to a specific type requires comparing its general term to the defining form. Providing counterexamples or explanations clarifies why a series does or does not fit a category, aiding in understanding convergence properties.