Ch. 10 - Sequences and Infinite Series

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 10 - Sequences and Infinite Series

Problem 10.1.19

Chapter 10, Problem 10.1.19

13–20. Explicit formulas Write the first four terms of the sequence { aₙ }∞ₙ₌₁.
aₙ = 1 + sin(πn / 2)

Verified step by step guidance

Identify the given explicit formula for the sequence: $a_n = 1 + \sin\left(\frac{\pi n}{2}\right)$, where $n$ is a positive integer starting from 1.

Recall that to find the first four terms of the sequence, you need to substitute $n = 1, 2, 3, 4$ into the formula one by one.

Calculate each term by plugging in the values of $n$: - For $n=1$, compute $a_1 = 1 + \sin\left(\frac{\pi \times 1}{2}\right)$. - For $n=2$, compute $a_2 = 1 + \sin\left(\frac{\pi \times 2}{2}\right)$. - For $n=3$, compute $a_3 = 1 + \sin\left(\frac{\pi \times 3}{2}\right)$. - For $n=4$, compute $a_4 = 1 + \sin\left(\frac{\pi \times 4}{2}\right)$.

Use your knowledge of sine values at special angles (multiples of $\frac{\pi}{2}$) to simplify each sine term without a calculator, for example, $\sin\left(\frac{\pi}{2}\right)$, $\sin(\pi)$, $\sin\left(\frac{3\pi}{2}\right)$, and $\sin(2\pi)$.

Write down the first four terms $a_1, a_2, a_3, a_4$ explicitly after simplification to complete the sequence.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule or formula. Each number in the sequence is called a term, denoted as aₙ, where n indicates the position. Understanding how to find terms from the formula is essential to write out the sequence.

Explicit Formula for Sequences

An explicit formula directly defines the nth term of a sequence as a function of n, allowing calculation of any term without knowing previous terms. For example, aₙ = 1 + sin(πn/2) gives a direct way to find each term by substituting n.

Evaluating Trigonometric Functions at Specific Angles

To find terms involving sine functions, you must evaluate sin(θ) at specific angles, often multiples of π. Knowing values like sin(0) = 0, sin(π/2) = 1, sin(π) = 0, and sin(3π/2) = -1 helps compute terms accurately.

Recommended video:

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Introduction to Trigonometric Functions

Related Practice

Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) ((3k³ + 4)(7k² + 1)) / ((2k³ + 1)(4k³ − 1))

Textbook Question

9–36. Comparison tests Use the Comparison Test or the Limit Comparison Test to determine whether the following series converge.

∑ (k = 1 to ∞) (2 + (−1)ᵏ) / k²

Textbook Question

16–17. {Use of Tech} Periodic savings

Suppose you deposit m dollars at the beginning of every month in a savings account that earns a monthly interest rate of r, which is the annual interest rate divided by 12 (for example, if the annual interest rate is 2.4%, r = 0.024/12 = 0.002). For an initial investment of m dollars, the amount of money in your account at the beginning of the second month is the sum of your second deposit and your initial deposit plus interest, or m + m(1 + r). Continuing in this fashion, it can be shown that the amount of money in your account after n months is

Aₙ = m + m(1 + r) + ⋯ + m(1 + r)ⁿ⁻¹.

Use geometric sums to determine the amount of money in your savings account after 5 years (60 months) using the given monthly deposit and interest rate.

17. Monthly deposits of \$250 at a monthly interest rate of 0.2%

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Textbook Question

Find a formula for the nth partial sum Sₙ of

∑ k = 1 to ∞[(1/(k + 3)) − (1/(k + 4))]

Use your formula to find the sum of the first 36 terms of the series.

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Textbook Question

72–86. Evaluating series Evaluate each series or state that it diverges.

∑ (k = 1 to ∞) (((1/6)ᵏ + (1/3)ᵏ) × k⁻¹)

Textbook Question

40–62. Choose your test Use the test of your choice to determine whether the following series converge.