Problem 39 - College Algebra

Question

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

Step-by-Step Solution

Answer

The sum of the series \(\sum_{i=0}^{4} \frac{(-1)^{i}}{i !} = 0.375\).

Step 1: Understand the Series Notation

The sum notation, represented by the Greek letter sigma (\(\Sigma\)), indicates that we are adding up the terms of a series. In this exercise, 'i' is the index of summation, and it varies from 0 to 4. For each value of 'i', the term to be added is computed as \((-1)^i / i!\). This notation tells us to plug each value from 0 to 4 into 'i' in the formula and then sum up those values.

Step 2: Calculate Individual Terms

Plug each integer from 0 to 4 into the formula. When \(i=0\), \((-1)^0 / 0! = 1/1 = 1\), when \(i=1\), \((-1)^1 / 1! = -1/1 = -1\),when \(i=2\), \((-1)^2 / 2! = 1/2 = 0.5\),when \(i=3\), \((-1)^3 / 3! = -1/6 \approx -0.167\), and finally, when \(i=4\), \((-1)^4 / 4! = 1/24 \approx 0.042\).

Step 3: Sum Up the Terms

Add up all the values computed in the previous step. So the final sum is \(1 - 1 + 0.5 - 0.167 + 0.042 = 0.375\).