Problem 61 - Precalculus: Mathematics for Calculus

Question

Write the sum without using sigma notation. $$\sum_{k=1}^{4} k^{3}$$

Step-by-Step Solution

Answer

1 + 8 + 27 + 64

Step 1: Understand the Sigma Notation

The expression \( \sum_{k=1}^{4} k^{3} \) means that we need to calculate the sum of \( k^3 \) for each integer value of \( k \) from 1 to 4.

Step 2: Compute Individual Terms

We will work out each term in the sequence separately:- When \( k = 1 \), \( k^3 = 1^3 = 1 \).- When \( k = 2 \), \( k^3 = 2^3 = 8 \).- When \( k = 3 \), \( k^3 = 3^3 = 27 \).- When \( k = 4 \), \( k^3 = 4^3 = 64 \).

Step 3: Write Without Sigma

Replacing the sigma notation with its expanded form, we write the sum without the sigma notation: \(1 + 8 + 27 + 64\).

Step 4: Calculate the Sum

We add the individual terms together: \(1 + 8 + 27 + 64 = 100\).

Key Concepts

Sum of Cubes

The concept of the sum of cubes is intriguing and presents a powerful mathematical pattern. The expression you often encounter, like \( \sum_{k=1}^{n} k^3 \), represents the sum of cubes of integers starting from 1 to \( n \). Each term, \( k^3 \), is the cube of an integer value \( k \). In simple terms, it tells you to take a number, multiply it by itself twice (like squaring but with an extra step), and you have the cube. For example, if you cube 2, you get \( 2 \times 2 \times 2 = 8 \). So, to find \( \sum_{k=1}^{4} k^3 \), you compute \( 1^3 + 2^3 + 3^3 + 4^3 \), which gives you 100.

• 1 cubed is 1
• 2 cubed is 8
• 3 cubed is 27
• 4 cubed is 64

Adding all this up gives the total of 100. This sum can have interesting applications and reveals patterns that can be generalized for any \( n \). It also connects to algebra through identities and can show why certain numbers form cubes.

Series Expansion

Series expansion is a natural mathematical tool used to express functions or sequences in a form that reveals the sums of individual terms. When dealing with a series, like the sum of cubes, the process involves breaking down the sigma (\( \Sigma \)) notation into a sequence that involves the explicit addition of each term.
This concept is key when working with sigma notation to uncover what each element or stage of the sequence equates to. Consider this exercise: you start with the sigma notation \( \sum_{k=1}^{4} k^3 \). By expanding it, you display it as \( 1 + 8 + 27 + 64 \).

• The process moves from a compact sigma notation to a detailed and expanded view of the series.
• Each number represents a term generated from cubing integers sequentially.
• These integers span from the lower to the upper bounds specified in the notation, here from 1 to 4.

This expansion is not just useful but central in further understanding sequences, calculus series, and various mathematical methods.

Mathematical Notation

Mathematical notation is the universal language of mathematics, encapsulating complex ideas succinctly and accurately. The use of symbols, like summation notation (\( \Sigma \)), turns lengthy processes into concise statements. Such notations are pivotal in simplifying and communicating intricate mathematical operations.
In this context, the notation \( \sum_{k=1}^{4} k^3 \) stands for the sum of cubes from 1 to 4. Breaking down the notation:

• The symbol \( \Sigma \) signifies summation.
• \( k \) is our index of summation, essentially a placeholder that takes on each integer value from 1 to 4.
• \( k^3 \) is the expression we evaluate for each \( k \).

Understanding and translating this notation allows students to avoid straightforward yet cumbersome arithmetic steps and explore deeper mathematical interpretations, improving both comprehension and analytical skills. It underscores the efficiency of mathematical language in diverse applications, from basic arithmetic to advanced computational theories.