In Exercises 29–42, find each indicated sum. 4Σi=0 (−1)^i/i!

Verified step by step guidance

1. Identify the components of the summation: The expression is a summation from i = 0 to 4 of the term (-1)^i / i!.
2. Understand the factorial notation: i! (i factorial) means the product of all positive integers up to i. For example, 0! = 1, 1! = 1, 2! = 2, 3! = 6, and 4! = 24.
3. Evaluate each term in the summation: Calculate each term (-1)^i / i! for i = 0, 1, 2, 3, and 4.
4. Sum the evaluated terms: Add the results of each term from i = 0 to 4 to find the total sum.
5. Review the process: Ensure each step is correctly followed and calculations are accurate to find the indicated sum.

Key Concepts

Summation Notation

Summation notation, represented by the sigma symbol (Σ), is a concise way to express the sum of a sequence of terms. In the given question, the notation 4Σi=0 indicates that we are summing terms from i=0 to i=4. Each term in the sum is defined by the expression (-1)^i/i!, which varies based on the value of i.

Factorial

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers up to n. For example, 3! = 3 × 2 × 1 = 6. In the context of the summation, the term i! in the denominator serves to scale the value of each term, affecting the overall sum as i increases.

Alternating Series

An alternating series is a series whose terms alternate in sign, such as the terms generated by (-1)^i. In this case, the series will have positive and negative terms based on the value of i, which influences the convergence and the final value of the sum. Understanding how these alternating signs affect the sum is crucial for evaluating the expression correctly.