Problem 102 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to find the sum. $$\sum_{k=0}^{4} \frac{(-1)^{k}}{k !}$$

Step-by-Step Solution

Answer

The sum of the series is 0.375.

Step 1: Understand the summation

The notation \( \sum_{k=0}^{4} \frac{(-1)^{k}}{k!} \) means that we are summing up a series from k=0 to k=4, where the series is defined as \(\frac{(-1)^{k}}{k!}\). In this series, the term \((-1)^k\) alternates in sign with each increase in k, and the term \(\frac{1}{k!}\) includes the factorial of k in the denominator. A factorial is the product of all positive integers less than or equal to that integer.

Step 2: Calculate each term

With the understanding of what each part of the series represents, we can then calculate each individual term in the series from k=0 to k=4: When k=0, the term would be \(\frac{(-1)^0}{0!} = 1\). When k=1, the term would be \(\frac{(-1)^1}{1!} = -1\). When k=2, the term would be \(\frac{(-1)^2}{2!} = \frac{1}{2}\). When k=3, the term would be \(\frac{(-1)^3}{3!} = -\frac{1}{6}\). And when k=4, the term would be \(\frac{(-1)^4}{4!} = \frac{1}{24}\).

Step 3: Sum up the terms

We then sum up all individual terms that we calculated in the previous step. The overall sum of the series is \(1 - 1 + 0.5 - \frac{1}{6} + \frac{1}{24} = 0.375\).