Problem 1 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 1 through 10, write the sum using summation notation. $$ 2^{3}+3^{3}+4^{3}+\cdots+19^{3} $$

Step-by-Step Solution

Answer

The summation notation of the series \(2^{3}+3^{3}+4^{3}+\cdots+19^{3}\) is \(\sum_{n=2}^{19} n^3\).

Step 1 - Identify the start and end of the sequence

The first number in the series is 2 and the last number is 19, which gives us the lower limit and upper limit of summation. This means \(n\) ranges from 2 to 19.

Step 2 - Express the sequence in summation notation

Each term in the series can be expressed as \(n^3\), where \(n\) is the term number starting from 2. Hence, the entire series can be expressed in summation notation as \(\sum_{n=2}^{19} n^3\).

Key Concepts

Series and Sequences

Understanding series and sequences is critical to various fields of mathematics, especially in calculus and analysis. A sequence is a list of numbers following a certain rule, where each number is called a term. Sequences can be finite or infinite, depending on whether they have a last term or not. In contrast, a series is the sum of the terms of a sequence. This is where we transition from merely listing numbers to adding them up, potentially to find their total or limit.

For instance, in the exercise provided, we encounter a finite sequence of the cubes of integers from 2 to 19. This sequence is finite because it has a clearly defined last term. When the terms of this sequence are added together, we form a series. In this case, the series is the sum of the cubes of all integers from 2 to 19. Recognizing the underlying sequence is the first step towards using summation notation to express the series, as was done in Step 1 of the problem's solution.

Expressing Series With Summation

Summation notation is a concise way to represent the sum of a series, particularly when dealing with a large number of terms or when the pattern of the sequence is clear. The sigma symbol \( \Sigma \) is used to denote summation and is accompanied by the expression for the terms of the series. It also involves specifying the variable that represents the index of terms, usually represented by \( n \) or \( i \) or other letters, starting from the lower limit and continuing to the upper limit of the series.

Considering the solution provided, \( \sum_{n=2}^{19} n^3 \) is the summation notation that concisely expresses the series of cubed numbers starting from 2 and ending at 19. This notation tells us to sum each term \( n^3 \) where \( n \) starts at 2 and increments by 1 until reaching the upper limit which is 19. Utilizing this powerful tool, we can express complex series with a simple and elegant notation.

Upper and Lower Limits of Summation

The limits of summation define the starting point (lower limit) and ending point (upper limit) for the sum. These limits are critical because they tell us the range over which we need to add up the terms of the sequence. The lower limit is usually written below the summation symbol, and the upper limit is written above.

In the exercise provided, the lower limit is 2 and the upper limit is 19. These are explicitly chosen based on the smallest and largest terms in the sequence that will be included in the series. These limits drive the value of the index \( n \)—which starts at 2 and increases by 1 with each term, stopping after the term for \( n=19 \) is added. It's important to accurately identify these limits to correctly express the summation and ensure that no terms are left out or erroneously included.