In Exercises 29–42, find each indicated sum. 5Σi=1 i!/(i−1)!
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factorial Notation
Factorial notation, denoted as n!, represents the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are fundamental in combinatorics and are used in various mathematical contexts, including permutations and series.
Summation notation, represented by the symbol Σ, is a concise way to express the sum of a sequence of terms. The notation Σi=1^n a_i indicates that you sum the terms a_i from i=1 to n. Understanding how to manipulate and evaluate sums is crucial for solving problems involving series.
Simplifying factorial expressions involves recognizing relationships between factorials. For instance, the expression i!/(i-1)! simplifies to i, since i! = i × (i-1)!. This simplification is essential for evaluating sums involving factorials, as it allows for easier computation and understanding of the series.