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Ch. 10 - Sequences and Infinite Series
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 10 - Sequences and Infinite SeriesProblem 10.1.10
Chapter 10, Problem 10.1.10

Given the series ∑∞ₖ₌₁ k, evaluate the first four terms of its sequence of partial sums Sₙ = ∑ⁿₖ₌₁ k. 

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1
Identify the given series: \( \sum_{k=1}^{\infty} k \) is the sum of natural numbers starting from 1.
Understand that the sequence of partial sums \( S_n \) is defined as \( S_n = \sum_{k=1}^n k \), which means adding the first \( n \) terms of the series.
Calculate the first partial sum \( S_1 \) by summing the first term: \( S_1 = 1 \).
Calculate the second partial sum \( S_2 \) by summing the first two terms: \( S_2 = 1 + 2 \).
Calculate the third and fourth partial sums similarly: \( S_3 = 1 + 2 + 3 \) and \( S_4 = 1 + 2 + 3 + 4 \).

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Key Concepts

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

Infinite Series and Partial Sums

An infinite series is the sum of infinitely many terms. The sequence of partial sums, Sₙ, represents the sum of the first n terms of the series. Evaluating partial sums helps understand the behavior of the series, especially whether it converges or diverges.
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Intro to Series: Partial Sums

Arithmetic Series

An arithmetic series is a sum of terms with a constant difference between consecutive terms. For the series ∑k, the terms increase by 1 each time. The sum of the first n terms can be found using the formula Sₙ = n(n + 1)/2.
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Geometric Series

Summation Notation and Indexing

Summation notation (∑) compactly represents the sum of terms indexed by k. Understanding how to interpret and manipulate the index and limits is essential for correctly evaluating partial sums and applying formulas.
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Related Practice
Textbook Question

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

∑ (k = 1 to ∞) 1 / k^(1 + p),p > 0

Textbook Question

Series of squares Prove that if ∑aₖ is a convergent series of positive terms, then the series ∑aₖ² also converges.

Textbook Question

48–63. Choose your test Determine whether the following series converge or diverge using the properties and tests introduced in Sections 10.3 and 10.4.

∑ (k = 1 to ∞) (5 / 6)⁻ᵏ

Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 10 to ∞) 1 / (k − 9)⁵

Textbook Question

13–52. Limits of sequences

Find the limit of the following sequences or determine that the sequence diverges.


{tan⁻¹(10n⁄(10n + 4))}  

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Textbook Question

1–10. Choosing convergence tests Identify a convergence test for each series. If necessary, explain how to simplify or rewrite the series before applying the convergence test. You do not need to carry out the convergence test.

∑ (from k = 3 to ∞) (2k²) / (k² − k − 2)