Problem 12 - Essential Calculus

Question

In Exercises 7 -12, use sigma notation to write the sum. $$ \left[1-\left(\frac{2}{n}-1\right)^{2}\right]\left(\frac{2}{n}\right)+\cdots+\left[1-\left(\frac{2 n}{n}-1\right)^{2}\right]\left(\frac{2}{n}\right) $$

Step-by-Step Solution

Answer

The given sum in sigma notation is \(\Sigma_{i=1}^{n} [1-\left(\frac{2i}{n}-1\right)^{2}]*\left(\frac{2i}{n}\right)\)

Step 1: Identify the components of the summation

Analyse the base pattern of the sum. The base pattern is \(1-\left(\frac{2i}{n}-1\right)^{2}\)*\(\frac{2i}{n}\) where \(i\) represents the term number in the sequence.

Step 2: Confirm the starting and ending values

The starting is \(i=1\) because the first term is using 1 in the place of \(i\). The ending value is \(n\) because in the final term, the expression \(\frac{2n}{n}\) indicates that \(n\) is replaced by \(n\).

Step 3: Write in sigma notation

Use the sigma notation with the identified components. The given sum can be written as \(\Sigma_{i=1}^{n} [1-\left(\frac{2i}{n}-1\right)^{2}]\left(\frac{2i}{n}\right)\). The index \(i\) starts at 1 and ends at \(n\), and each term of the sum has the form \([1-(\frac{2i}{n}-1)^{2}]\)*(\frac{2i}{n})\).

Key Concepts

Summation

Summation is the process of adding a sequence of numbers or expressions. It is an essential concept in mathematics, often used to simplify the process of adding a large number of terms. Instead of writing out each addition individually, summation allows for a compact representation of the process.

The sum is denoted by the Greek letter sigma, written as \( \Sigma \). Following sigma, there is an expression defining the terms to be added. This expression typically involves a variable, often denoted by \( i \) or \( n \), which takes on integer values over a specified range.

The power of sigma notation lies in its ability to succinctly express large sums. It is especially useful in calculus for expressing series and in statistics for expressing the sum of data points.

Sigma Notation Form

Sigma notation is a concise way to represent the sum of a sequence of terms. Understanding its form is key to interpreting and creating expressions with it. A standard sigma notation looks like \( \Sigma_{i=a}^{b} f(i) \), where:

• \( \Sigma \) is the summation symbol, indicating the operation of addition.
• \( i \) is the index of summation; it takes on integer values starting from \( a \) to \( b \).
• \( f(i) \) is the function or expression whose values are being added; it depends on the index \( i \).
• \( a \) is the starting value for \( i \), and \( b \) is the ending value.

This notation compactly represents adding up terms starting from when \( i = a \) to when \( i = b \). Each term is evaluated and then summed. For our example, the expression is written as \( \Sigma_{i=1}^{n} \left[1-\left(\frac{2i}{n}-1\right)^{2}\right]\left(\frac{2i}{n}\right) \), where each component has a distinct role in forming each term of the sum.

Sequence and Series

Before diving into sigma notation, it's crucial to understand sequences and series, as they form the foundation of summation.

A *sequence* is an ordered list of numbers following a particular pattern or rule. Each number is termed an element of the sequence. Sequences can be finite or infinite. For example, \( \{1, 4, 7, 10, \ldots\} \) is a sequence where each term increases by 3.

A *series* is what results when all elements in a sequence are summed together. If the sequence is finite, the series will yield a single sum. If it's infinite, methods of analysis help assess its convergence, or finiteness.

In the exercise given, each term in the sequence is generated by the expression \( \left[1-\left(\frac{2i}{n}-1\right)^{2}\right]\left(\frac{2i}{n}\right) \), signifying a potentially complex pattern of numbers. Summing these terms using sigma notation allows encapsulating the entire sequence as a series that can be computed for a chosen \( n \).