Problem 41 - Precalculus Mathematics for Calculus

Question

Find the sum. $$\sum_{k=1}^{4} k$$

Step-by-Step Solution

Answer

10

Step 1: Understand the Summation Notation

The expression \( \sum_{k=1}^{4} k \) represents the sum of a series. The index \( k \) starts at 1 and increases by 1 until it reaches 4. We need to add all of the values of \( k \) that satisfy this condition.

Step 2: Write Out the Series

Convert the series into a list of terms. For \( \sum_{k=1}^{4} k \), we write the series as \( 1 + 2 + 3 + 4 \).

Step 3: Calculate the Sum

Simply add the numbers from the series: \( 1 + 2 + 3 + 4 = 10 \). This is the sum of the series.

Key Concepts

Series

In mathematics, a **series** is the sum of the terms of a sequence. To understand it better, think of a sequence as a list of numbers arranged in a specific order, like 1, 2, 3, 4, and so on. A series is what you get when you add up these numbers. For example, in the sequence 1, 2, 3, and 4, the corresponding series is their sum, which is 1 + 2 + 3 + 4. Let's break it down:

• A series always involves addition.
• The numbers in a series come from a sequence.
• Series can be finite (with a limited number of terms) or infinite.

In the context of our exercise, the series comes from the sequence formed by the numbers in the range from 1 to 4. We use summation notation (the big capital sigma symbol) to efficiently represent long series without writing all the numbers out. It does the job of neatly packaging the sequence and its sum all in one place.

Arithmetic Series

An **arithmetic series** is a type of series where each term in the sequence increases by a constant amount, known as the common difference. In simpler words, you keep adding the same number to each term to get to the next.Here's a quick guide:

• An arithmetic series starts with a first term.
• Each following term is obtained by adding a fixed number, called the common difference.
• You can have both finite and infinite arithmetic series, but finite ones are easier to calculate.

The exercise we've solved is a perfect example of a finite arithmetic series where the common difference is 1. Starting from 1, you add 1 repeatedly to get to 2, 3, and finally 4. The series \(1 + 2 + 3 + 4 \) results from this process. In general, the sum of the first \(n\) terms of an arithmetic series can be given by a formula, but since our series is short, simple addition is more practical.

Index Notation

**Index notation**, often seen using the summation symbol \( \Sigma \), is a compact way to express the sum of terms that follow a particular pattern. This notation is especially useful in mathematics for dealing with long series effortlessly.Here’s how it works:

• The lower part of the sigma symbol tells us where to start in the sequence (e.g., \(k=1\)).
• The upper part tells us where to end (e.g., \(4\)).
• It allows us to define the rule for generating terms (e.g., the term is just \(k\) itself in \(\sum_{k=1}^{4} k\)).

Consider the example from our exercise: \(\sum_{k=1}^{4} k\). Here, the index \(k\) takes the values 1, 2, 3, and 4. For each value, it simply represents itself, so we write the series as \(1 + 2 + 3 + 4\). Using index notation helps you quickly set up and solve summation problems with clarity and efficiency.