Problem 12 - Thomas Calculus

Question

Express the sums in Exercises \(11-16\) in sigma notation. The form of your answer will depend on your choice of the lower limit of summation. $$ 1+4+9+16 $$

Step-by-Step Solution

Answer

The sum is \(\sum_{n=1}^{4} n^2\).

Step 1: Identify the Pattern

Look at the sequence given: \(1, 4, 9, 16\). Notice that each term can be expressed as a perfect square, specifically: \(1^2, 2^2, 3^2, 4^2\). Thus, the \(n\)-th term of the sequence is \(n^2\).

Step 2: Define the Index of Summation

In sigma notation, we need an index to denote each term's position in the series. We will use \(n\) as our index, which will represent each term's square value. For our series \(1^2, 2^2, 3^2, 4^2\), \(n\) will run from 1 to 4.

Step 3: Write in Sigma Notation

Using the pattern and the index defined, the sum can be written in sigma notation as \(\sum_{n=1}^{4} n^2\). This notation corresponds to summing the squares of \(n\) from 1 to 4.

Key Concepts

Infinite Series

An infinite series is a sum of an unending sequence of terms. However, the series in our exercise is not infinite because it has a definite number of terms. Instead, this series is an example of a finite sequence, consisting of numbers like 1, 4, 9, and 16. In mathematics, however, infinite series are quite significant. They are like their finite counterparts but extend indefinitely. This implies that the summation has no end, like the sum of all natural numbers or all the reciprocals. These sums often converge at a particular value; that is, no matter how long you continue to add the terms, the sum approaches a specific number.

For instance, the sum of 1/n where n goes from 1 to infinity is known as the harmonic series. It diverges, which means it does not converge to a particular number as you keep adding more terms.

• Infinite series often help solve complex problems in calculus and are fundamental in defining many mathematical constants.
• Understanding them requires a good grasp of limits, which show how a series behaves as more terms are added.

Summation Index

The summation index is a term used to represent the position of each term in a series. It is denoted by symbols such as \( n \), \( i \), or \( k \), and it serves as a placeholder in sigma notation. In our exercise, the index \( n \) represents the exponent values, indicating each term in the sequence as a perfect square.

When writing in sigma notation, you will see expressions like \( \sum_{n=1}^{4} n^2 \). Here, "n" is the summation index that takes values from the lower limit, 1, up to the upper limit, 4. This tells us to sum up all the values that fit the pattern \( n^2 \) as \( n \) moves through the integers from 1 to 4.

• The lower limit of the index shows the starting point of the series.
• The upper limit indicates where the series ends.
• It simplifies the way we write long sums, making expressions more compact and understandable.

Perfect Squares

Perfect squares are numbers that result from squaring a whole number. In simpler terms, a perfect square is the product of an integer with itself. Numbers such as 1, 4, 9, and 16 are examples because they are 1 squared (\( 1^2 \)), 2 squared (\( 2^2 \)), 3 squared (\( 3^2 \)), and 4 squared (\( 4^2 \)), respectively.

The exercise's sequence forms perfect squares, leading us to use the pattern \( n^2 \) where "n" is an integer representing the term's position.

• Recognizing perfect squares is a key skill, helping to simplify problems and make connections in algebra and number theory.
• They frequently appear in various mathematical contexts, including the Pythagorean theorem and quadratic equations.
• Spotting these can also assist in solving math problems efficiently, as seen in creating sigma notation for a given sequence.