Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problema 10.4.47a

Capitolo 10, Problema 10.4.47a

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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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.

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P-Series and Harmonic 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.

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Percorso guidato

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Geometric 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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Percorso guidato

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Geometric Series

Pratica correlata

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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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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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87. Explain why or why not

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

a. If ∑ (k = 1 to ∞) aₖ converges, then ∑ (k = 10 to ∞) aₖ converges.

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