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Chapter 6: Problem 67

Evaluate the series. $$\sum_{k=1}^{4} \frac{1}{k}$$

Short Answer

Expert verified
The sum of the series \(\sum_{k=1}^{4} \frac{1}{k}\) is approximately 2.083

Step by step solution

01

Identify the series type

The given series is of the form \(\sum_{k=1}^{4} \frac{1}{k}\), which is a harmonic series. In a harmonic series, each term is the reciprocal of a positive integer.
02

Calculate each term of the series

In order to calculate the sum of the series, evaluate each term in the series:For \(k = 1\), \(\frac{1}{k} = 1\),For \(k = 2\), \(\frac{1}{k} = 0.5\),For \(k = 3\), \(\frac{1}{k} = \frac{1}{3}\) = approximately 0.333,For \(k = 4\), \(\frac{1}{k} = 0.25\).
03

Add all terms together

Now, add all the terms together to find the sum of the series. \(\sum_{k=1}^{4} \frac{1}{k} = 1 + 0.5 + 0.333 + 0.25 = 2.083 \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series Evaluation
When evaluating a series, it is crucial to first recognize the type of series you are dealing with. In the provided problem, we have a harmonic series. To evaluate a series:
  • Identify the formula or pattern each term follows.
  • Determine the value or range of values of the index involved, here it is denoted by "k" ranging from 1 to 4.
  • Calculate each term in the series using the given formula.
  • Add up all calculated terms to find the total sum of the series.
This step-by-step evaluation is not just about calculating numbers; it helps build a deeper understanding of sequences and the roles they play within series.
Reciprocal
A reciprocal refers to the multiplicative inverse of a number. Simply put, for a fraction or a number "x," the reciprocal is calculated as \(\frac{1}{x}\). In this exercise, each term of the series is the reciprocal of a positive integer. For example, the reciprocal of 1 is \(\frac{1}{1} = 1\), 2 is \(\frac{1}{2} = 0.5\), and so on.

Understanding reciprocals is fundamental in solving harmonic series, as they form the backbone of calculating each term:
  • Recognize that a reciprocal effectively translates an integer to its "one-over" form, crucial for harmonic series.
  • Use reciprocals to transform any integer into a fraction, which is in turn summed up over the series.
Thus, reciprocals serve as a key concept in the straightforward evaluation of the series discussed.
Sum Series
A sum series involves adding the values of terms in the sequence over a specified range. In the case of our exercise, the range is from 1 to 4, and each term is expressed as the reciprocal of its index number.

To find the sum of our series:
  • Calculate each term, which we've done previously as 1, 0.5, \(\frac{1}{3}\), and 0.25.
  • Add these calculated terms together.
  • The final addition, \(1 + 0.5 + \frac{1}{3} + 0.25\), gives the approximate sum of 2.083.
The sum not only provides a numerical result but also reflects the process of evaluating the pattern and progression of terms within a sequence. This methodology is applicable in various mathematical scenarios involving sequences and series.
Sequence and Series
Considering both sequences and series is essential to understanding their structure and evaluation. A sequence refers to a list of numbers following a particular pattern, while a series is the sum of such a sequence.

In the exercise example, the sequence \(a_1, a_2, a_3, a_4\) is given by \(1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}\). This sequence is the base of the series.
  • Recognize the sequential pattern of the terms, here as the reciprocals of integers.
  • Convert the sequence into a series by summing the terms according to the specified index range.
Sequence and series concepts are interrelated, where understanding one helps in the comprehensive evaluation and sum of the other. This insight fosters improved problem-solving skills in mathematics.

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