Textbook Question
67–70. Formulas for sequences of partial sums Consider the following infinite series.
c.Make a conjecture for the value of the series.
∑⁽∞⁾ₖ₌₁2⁄[(2k − 1)(2k + 1)]
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14. Sequences & Series
Series
Problem 10.4.47b
Textbook Question
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 guidance1
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.
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 \(k=3\), \(j=1\).
Express the series in terms of \(j\): \(\sum_{j=1}^{\infty} \frac{1}{\sqrt{j}} = \sum_{j=1}^{\infty} \frac{1}{j^{1/2}}\).
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 \(p = \frac{1}{2}\).
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
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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P-Series and Harmonic Series
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.
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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.
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04:30P-Series and Harmonic Series
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