Problem 39 - Algebra and Trigonometry

Question

find each indicated sum. $$ \sum_{i=0}^{4} \frac{(-1)^{i}}{i !} $$

Step-by-Step Solution

Answer

The sum of the series is \(\frac{1}{3}\).

Step 1: Understand the Sigma Notation

The sigma notation, denoted by \(\sum\), indicates that we need to add together the terms of a sequence. The variable 'i' is our index of summation, which ranges from 0 to 4 in this case. We sum the terms of the sequence defined by the formula \(\frac{(-1)^{i}}{i !}\) for each value of 'i' in that range.

Step 2: Calculate Individual Terms

We substitute 0, 1, 2, 3, and 4 into the formula to get the terms of the sequence:When \(i=0\), the term is \(\frac{(-1)^{0}}{0 !} = 1\).When \(i=1\), the term is \(\frac{(-1)^{1}}{1 !} = -1\).When \(i=2\), the term is \(\frac{(-1)^{2}}{2 !} = \frac{1}{2}\).When \(i=3\), the term is \(\frac{(-1)^{3}}{3 !} = -\frac{1}{6}\).When \(i=4\), the term is \(\frac{(-1)^{4}}{4 !} = \frac{1}{24}\).

Step 3: Add the Terms Together

We add the sequence's terms together to get the sum of the series:\(1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} = \frac{1}{3}\).

Key Concepts

Summation Index

The summation index is an essential part of the sigma notation used for summing series. In our example, the variable 'i' is the summation index. It tells us which numbers should be plugged into the expression to compute the sequence terms.
The index of summation indicates the scope or range for the sum, usually positioned directly under the sigma symbol.
In the given problem, the summation index is 'i', and it ranges from 0 to 4. This means we substitute each integer in this range into the given formula to calculate each term of the sequence: \( \frac{(-1)^{i}}{i !} \).

• The summation index starts at 0 (the lower bound) and goes up sequentially to the upper bound, which is 4.
• Once each term is calculated with the index, they will all be added together to get the final series sum.

Understanding the summation index is crucial for correctly calculating each term and ultimately finding the sum of the series.

Factorial

In mathematics, factorial is a function that multiplies a series of descending natural numbers. The notation for factorial is the exclamation point '!'. For a positive integer 'n', \( n! \) is the product of all positive integers less than or equal to n.
For the special case of 0, \( 0! \) is defined to be 1, which can sometimes be a surprising rule for beginners.
So, how does this relate to our original problem? When handling the formula \( \frac{(-1)^{i}}{i !} \), you'll compute the factorial of each 'i' value:

• \(0! = 1\)
• \(1! = 1\)
• \(2! = 2 \times 1 = 2\)
• \(3! = 3 \times 2 \times 1 = 6\)
• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

This factorial calculation helps you simplify the expression inside the sigma notation, allowing you to calculate each term of the sequence accurately.

Sequence Terms

The terms of a sequence are the individual elements that get added together when performing a summation. Each term is calculated using the given formula and the range specified by the summation index.
In our exercise, the formula provided is \( \frac{(-1)^{i}}{i !} \). With each value of 'i', a different sequence term is generated:

• When \( i = 0 \), \( \frac{(-1)^{0}}{0!} = 1 \)
• When \( i = 1 \), \( \frac{(-1)^{1}}{1!} = -1 \)
• When \( i = 2 \), \( \frac{(-1)^{2}}{2!} = \frac{1}{2} \)
• When \( i = 3 \), \( \frac{(-1)^{3}}{3!} = -\frac{1}{6} \)
• When \( i = 4 \), \( \frac{(-1)^{4}}{4!} = \frac{1}{24} \)

Each calculated term contributes to finding the total value of the series sum. The signs alternate because of the \((-1)^{i}\) component, introducing negative terms based on whether 'i' is odd or even.

Series Sum

The series sum is the result of adding up all the terms calculated from a sequence. A series can be thought of as the final result after applying the sigma notation.
In our example, after finding individual sequence terms for \( i = 0 \) to \( i = 4 \), we add them together. These terms are:

• \(1\)
• \(-1\)
• \(\frac{1}{2}\)
• \(-\frac{1}{6}\)
• \(\frac{1}{24}\)

When summed: \( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} \), it equals \( \frac{1}{3} \).
This final value, \( \frac{1}{3} \), is the series sum, representing the culmination of the summation process for this specific problem. Understanding the transition from individual terms to a comprehensive sum is key to mastering series and sigma notation in mathematics.