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Chapter 14: Problem 27

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

Short Answer

Expert verified
\(\frac{2}{3}\).

Step by step solution

01

Write out the Sequence

Let's list each term of the sequence for \(i=0\) to \(i=4\): \[0th term: \frac{(-1)^{0}}{0 !},\] \[1st term: \frac{(-1)^{1}}{1 !},\] \[2nd term: \frac{(-1)^{2}}{2 !},\] \[3rd term: \frac{(-1)^{3}}{3 !},\] \[4th term: \frac{(-1)^{4}}{4 !}\].
02

Simplify Each Term

Now, let's simplify each term: \[0th term: \frac{(-1)^{0}}{0 !}=1,\] \[1st term: \frac{(-1)^{1}}{1 !}=-1,\] \[2nd term: \frac{(-1)^{2}}{2 !}=1/2,\] \[3rd term: \frac{(-1)^{3}}{3 !}=-1/6,\] \[4th term: \frac{(-1)^{4}}{4 !}=1/24\].
03

Add the Sequence Terms Together

Now add all the terms together: \[1+(-1)+\frac{1}{2}+(-\frac{1}{6})+\frac{1}{24} = \frac{2}{3}\].

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I was able to find the sum of the first fifty terms of an arithmetic sequence even though I did not identify every term.

Use the formula for the sum of the first n terms of a geometric sequence to solve. You save \(\$ 1\) the first day of a month, \(\$ 2\) the second day, \(\$ 4\) the third day, continuing to double your savings each day. What will your total savings be for the first 15 days?

Explain how to find \(n !\) if \(n\) is a positive integer.

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

What is the common ratio in a geometric sequence?
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