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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 5 - IntegrationProblem 5.1.47d
Chapter 5, Problem 5.1.47d

Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(d) 1 + 1/2 + 1/3 + 1/4

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1
Identify the pattern in the given sequence: The terms are fractions with the numerator fixed at 1 and the denominator increasing sequentially (1, 2, 3, 4).
Recognize that the general term for this sequence can be expressed as 1k, where k represents the position of the term in the sequence.
Determine the range of the index k: The sequence starts at k=1 and ends at k=4.
Write the sum in sigma notation: ∑k1^41k, where the summation symbol ∑ indicates the sum of the terms from k=1 to k=4.
Verify the sigma notation by expanding it: Substitute k values (1, 2, 3, 4) into the general term 1k to confirm that it matches the original sequence.

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Key Concepts

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

Sigma Notation

Sigma notation is a concise way to represent the sum of a sequence of terms. It uses the Greek letter sigma (Σ) to indicate summation, followed by an expression that defines the terms to be added. The notation typically includes an index of summation, which specifies the starting and ending values for the variable that represents the terms.
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Sigma Notation

Index of Summation

The index of summation is a variable used in sigma notation to denote the position of each term in the sequence being summed. It usually starts at a specified lower limit and increments by one until it reaches an upper limit. For example, in the sum Σ from i=1 to n, 'i' is the index that takes on values from 1 to n.
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Sigma Notation

Harmonic Series

The series represented by the sum 1 + 1/2 + 1/3 + 1/4 is known as the harmonic series. Each term in this series is the reciprocal of a positive integer. Understanding the harmonic series is important because it illustrates concepts of convergence and divergence in infinite series, although in this case, we are only summing a finite number of terms.
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P-Series and Harmonic Series
Related Practice
Textbook Question

Properties of integrals Consider two functions ƒ and g on [1,6] such that ∫₁⁶ƒ(𝓍) d𝓍 = 10 and ∫₁⁶g(𝓍) d𝓍 = 5, ∫₄⁶ƒ(𝓍) d𝓍 = 5 , and ∫₁⁴g(𝓍) d𝓍 = 2. Evaluate the following integrals.


(d) ∫₄⁶ (g(𝓍) ― f(𝓍) d𝓍

Textbook Question

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.


ƒ(𝓍) = 2x + 1 on [0,4] ; n = 4


d) Calculate the midpoint Riemann sum.

Textbook Question

Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.



Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n. 

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral..


∫₀² (𝓍²―2) d𝓍 ; n = 4

Textbook Question

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₋₂ˣ ƒ(t) dt and F(x) = ∫₄ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(4)

Textbook Question

Midpoint Riemann sums Complete the following steps for the given function, interval, and value of n.


ƒ(𝓍) = 1/x on [1,6] ; n = 5


(d) Calculate the midpoint Riemann sum.