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


b. The sum ∑ (k = 3 to ∞) 1 / √(k − 2) 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 constant.
2. Rewrite the given series \(\sum_{k=3}^{\infty} \frac{1}{\sqrt{k - 2}}\) by making a substitution to see if it matches the p-series form. Let \(j = k - 2\), so when \(k=3\), \(j=1\).
3. Express the series in terms of \(j\): \(\sum_{j=1}^{\infty} \frac{1}{\sqrt{j}} = \sum_{j=1}^{\infty} \frac{1}{j^{1/2}}\).
4. Since the series can be written as \(\sum_{j=1}^{\infty} \frac{1}{j^{1/2}}\), it matches the form of a p-series with \(p = \frac{1}{2}\).
5. Therefore, the given series is a p-series because it can be expressed in the form \(\sum \frac{1}{k^p}\) with \(p = \frac{1}{2}\).

Key Concepts

Definition of a p-series

A p-series is an infinite series of the form ∑ 1/n^p, where n starts from 1 or another positive integer, and p is a positive real number. The behavior and convergence of the series depend on the value of p.

Index shift in series notation

Changing the index of summation or shifting the variable inside the series can transform the series into a more recognizable form. Understanding how to rewrite sums by adjusting indices helps identify the type of series.

Criteria for identifying p-series

To determine if a series is a p-series, the general term must be expressible as 1/(n^p) for some p. If the term involves a shifted index but can be rewritten to fit this form, it qualifies as a p-series.