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.

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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.

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 $k$.

Since the denominator is not of the form $k^p$, the series does not fit the definition of a p-series.

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 $\frac{1}{3}$.

Therefore, the statement that the series is a p-series is false; it is actually a geometric series.

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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 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.

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Geometric Series

Related Practice

Textbook Question

88–89. Binary numbers

Humans use the ten digits 0 through 9 to form base-10 or decimal numbers, whereas computers calculate and store numbers internally as binary numbers—numbers consisting entirely of 0’s and 1’s. For this exercise, we consider binary numbers that have the form 0.b₁b₂b₃⋯, where each of the digits b₁, b₂, b₃, ⋯ is either 0 or 1. The base-10 representation of the binary number 0.b₁b₂b₃⋯ is the infinite series

b₁ / 2¹ + b₂ / 2² + b₃ / 2³ + ⋯

89. Computers can store only a finite number of digits and therefore numbers with nonterminating digits must be rounded or truncated before they can be used and stored by a computer.

a. Find the base-10 representation of the binary number 0.001̅1.

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Textbook Question

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a.If limₙ→∞aₙ = 1 and limₙ→∞bₙ = 3, then limₙ→∞(bₙ / aₙ) = 3.

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Textbook Question

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.

a. Find the next two terms of the sequence.

{-5, 5, -5, 5, ......}

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