/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 39 Find each indicated sum. $$\su... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 39

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

Short Answer

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

Step by step solution

01

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.
02

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 \(i=4\), \((-1)^4 / 4! = 1/24 \approx 0.042\).
03

Sum Up the Terms

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

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Most popular questions from this chapter

Find the term indicated in each expansion. \((x-1)^{9} ;\) fifth term

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Explain how to find the sum of the first \(n\) terms of a geometric sequence without having to add up all the terms.

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