Find each indicated sum. ${\sum }_{i=1}^5\frac{i!}{(i-1)!}$

Verified step by step guidance

1. First, understand the summation notation: \(\sum_{i=1}^{5} \frac{i!}{(i-1)!}\) means you will add the values of the expression \(\frac{i!}{(i-1)!}\) for each integer \(i\) from 1 to 5.
2. Next, simplify the general term \(\frac{i!}{(i-1)!}\). Recall that factorial notation means \(n! = n \times (n-1) \times (n-2) \times \cdots \times 1\), and specifically, \(i! = i \times (i-1)!\).
3. Using this, rewrite the term as \(\frac{i!}{(i-1)!} = \frac{i \times (i-1)!}{(i-1)!}\). Since \((i-1)!\) appears in both numerator and denominator, they cancel out, leaving you with \(i\).
4. Now, the summation simplifies to \(\sum_{i=1}^{5} i\), which means you need to add the integers from 1 to 5.
5. Finally, write out the sum explicitly as \$1 + 2 + 3 + 4 + 5$ and then add these values together to find the total sum.

Key Concepts

Summation Notation

Summation notation (Σ) is a concise way to represent the sum of a sequence of terms. The index variable (i) runs from the lower limit to the upper limit, and each term is evaluated at each integer value of i. Understanding how to interpret and expand summation notation is essential for solving series problems.

Factorials

A factorial, denoted by n!, is the product of all positive integers from 1 up to n. For example, 5! = 5×4×3×2×1 = 120. Factorials are used in permutations, combinations, and series expressions, and simplifying factorial expressions often involves canceling common terms.

Simplifying Factorial Expressions

When factorials appear in fractions, such as i!/(i−1)!, they can often be simplified by canceling common factors. Since i! = i × (i−1)!, the expression simplifies to i. Recognizing this simplification helps in evaluating sums more efficiently.