Problem 39 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The sum is 10.

Step 1: Identify the Given Series

The problem asks us to find the sum of a series, specified as \( \sum_{k=1}^{4} k \). This means we need to sum up all integers \( k \) from 1 to 4.

Step 2: Expand the Series

Expand the series into its individual terms. For \( \sum_{k=1}^{4} k \), the terms are: 1, 2, 3, and 4.

Step 3: Add the Series Terms

Add the expanded terms together: \[1 + 2 + 3 + 4 = 10\]

Step 4: Confirm the Result

Verify the addition: first, add 1 and 2 to get 3. Then, add 3 to get 6, and finally add 4 to get 10. This confirms that the sum is indeed correct.

Key Concepts

Series Expansion

When confronted with a series like \( \sum_{k=1}^{4} k \), the first step is to expand the series. Series expansion is a vital concept in mathematics that allows us to visualize each term we are summing. This particular series represents the set of consecutive integers from 1 to 4.

• Start by identifying each number in the sequence, from the initial value of \( k = 1 \) to \( k = 4 \).
• The terms you'll deal with are 1, 2, 3, and 4.
• By expanding the series, you get the individual components that will be added together.

As you can see, expanding the series breaks down the task into smaller, more manageable parts, making it easier to sum the numbers.

Summation Notation

Summation notation, denoted as \( \sum \), is a succinct way to represent the summation of a sequence of numbers. It is particularly useful in indicating a series to be summed over, which can greatly simplify complex calculations.

• In \( \sum_{k=1}^{4} k \), the Greek letter \( \Sigma \) (sigma) signifies the sum.
• The number beneath sigma, \( k=1 \), is the starting point of your summation.
• The upper number, 4, indicates the highest value of \( k \) included in your sum.
• The expression \( k \) represents the general term of the series.

Through this notation, you are informed quickly about what numbers are included without having to write each one down explicitly.

Finite Series

A finite series, like \( \sum_{k=1}^{4} k \), comprises a limited number of terms to be summed. This characteristic makes finite series simple to work with and understand for beginners.

• The term 'finite' means that the sequence has a clear beginning and endpoint, defined within the summation notation.
• In our example, the sequence starts at 1 and ends at 4.
• Finite series differ from infinite series that continue indefinitely, which require more complex analysis.

Finite series are commonly used in elementary mathematics to develop a foundational understanding of summation before moving on to more complex, infinite series concepts.