Problem 41 - Algebra and Trigonometry

Question

Find the sum. $$\sum_{k=1}^{3} \frac{1}{k}$$

Step-by-Step Solution

Answer

The sum is \( \frac{11}{6} \).

Step 1: Expand the sum

\( \sum_{k=1}^{3} \frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} = 1 + \frac{1}{2} + \frac{1}{3} \).

Step 2: Find common denominator

\( = \frac{6}{6} + \frac{3}{6} + \frac{2}{6} = \frac{11}{6} \).

Key Concepts

Sequence

In mathematics, a sequence is a list of numbers arranged in a specific order. Each number in the sequence is called a term. Sequences can be finite or infinite, depending on whether they have a specific number of terms or continue indefinitely. For example, the sequence for the exercise \( \frac{1}{1}, \frac{1}{2}, \frac{1}{3} \) is a finite sequence comprised of three terms. In this particular sequence:

• Each term is the reciprocal of its position in the list.
• The sequence begins with \( \frac{1}{1} \) and ends with \( \frac{1}{3} \).

Sequences can be described using a general formula. For this sequence, the formula for the \( k \)-th term is \( \frac{1}{k} \). This formula helps us find any term in the sequence based on its position \( k \). Recognizing the sequence allows you to better understand its structure and calculate sums or other characteristics.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence. However, the problem we looked at is actually not an arithmetic series, but rather a sum of fractions as noted in the sequence \( \frac{1}{1}, \frac{1}{2}, \frac{1}{3} \). Unlike an arithmetic series where the difference between each consecutive term is constant, this series consisted of fractions, where the terms are determined by their position \( k \) as \( \frac{1}{k} \).In any series, the concept remains the same, which is to find the sum of the sequence. By breaking down what each term in a sequence represents and calculating their cumulative sum, we can find the value of the summation notation, \( \sum_{k=1}^{3} \frac{1}{k} \). The notation encourages the methodical evaluation of each term in sequence to achieve the total sum.

Mathematical Notation

Mathematical notation provides a language for clearly communicating complex mathematical concepts. One powerful piece of notation is the summation symbol, \( \sum \), which simplifies expressing sums of sequences. A sum like \( \sum_{k=1}^{3} \frac{1}{k} \) uses the Greek letter Sigma \( \sum \) to indicate the sum over a sequence.Summation notation comprises:

• A starting index, here \( k = 1 \).
• An ending index, \( k = 3 \).
• The general term for the sequence, \( \frac{1}{k} \).

This notation helps organize complicated series expressions into a manageable format. It directs summing from \( k=1 \) to \( k=3 \) for all terms defined by \( \frac{1}{k} \). Mastering this notation is essential because it not only applies to arithmetic series but also broader mathematical scenarios. Mathematical notation succinctly conveys processes that would otherwise require extensive text.