Problem 3 - Essential Calculus

Question

In Exercises \(1-6,\) find the sum. Use the summation capabilities of a graphing utility to verify your result. $$ \sum_{k=0}^{4} \frac{1}{k^{2}+1} $$

Step-by-Step Solution

Answer

The sum of the series \(\sum_{k=0}^{4} \frac{1}{k^{2}+1}\) is \(\frac{351}{170}\)

Step 1: Identify the series

The series given is \(\sum_{k=0}^{4} \frac{1}{k^{2}+1}\). This means that we need to evaluate the expression \(\frac{1}{k^{2}+1}\) for each value of \(k\) from 0 to 4 and then add the results together.

Step 2: Substitute and calculate

Substitute each value of \(k\) from 0 through 4 into the formula. For \(k=0\), \(\frac{1}{0^{2}+1} = 1\), for \(k=1\), \(\frac{1}{1^{2}+1} = \frac{1}{2}\), for \(k=2\), \(\frac{1}{2^{2}+1} = \frac{1}{5}\), for \(k=3\), \(\frac{1}{3^{2}+1} = \frac{1}{10}\), and for \(k=4\), \(\frac{1}{4^{2}+1} = \frac{1}{17}\).

Step 3: Sum the results

Now, we add these results together. So, \(\sum_{k=0}^{4} \frac{1}{k^{2}+1} = 1 + \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} = \frac{351}{170}\).

Key Concepts

Summation Notation

The summation notation is a concise way to represent the addition of a sequence of numbers. This powerful mathematical tool, denoted by the Greek capital letter sigma ( Σ), allows us to express the sum of a series in compact form. In our exercise, we have the series Σ from k=0 to 4 of (1 / (k² + 1)). The lowercase 'k' represents the variable, 0 is the lower bound (the starting index), and 4 is the upper bound (the ending index). Summation notation requires us to evaluate the expression (1 / (k² + 1)) for every integer value of 'k' within the bounds, and then sum those values. It is a stepping-stone in learning to understand patterns and sequences in mathematics. While initially it may seem daunting, once understood, the summation notation provides a quick and systematic approach to adding up a series of terms.

Series Calculation

Calculating the sum of a series involves evaluating each individual term and adding them together. In this exercise, the series is finite and consists of five terms determined by substituting values of 'k' from 0 through 4 into the given formula, (1 / (k² + 1)). Once we substitute these values, we tally the results to find the total sum. This process requires careful calculation; each term needs to be accurately computed to ensure the final sum is correct. In some cases, especially with longer or more complex series, series calculation can become cumbersome. This is why having a solid grasp of the method and a keen attention to detail is fundamental in achieving accurate results. As a tip, it's always useful to double-check each term before summing them up to avoid any small errors that might lead to incorrect final answers.

Graphing Utility Verification

With advancements in technology, graphing utilities have become an invaluable resource for verifying mathematical calculations, especially when dealing with series and summation. Once the series is calculated by hand, a graphing utility can be used as a reliable method to check the work. By entering the series into the utility, it can compute the sum quickly, providing confirmation or catching any potential mistakes. Besides verification, a graphing utility can offer visual insights into the behavior of a series, such as its convergence or divergence. In our exercise, using a graphing utility to verify the sum of Σ from k=0 to 4 of (1 / (k² + 1)) not only reinforces the calculation but also increases confidence in the final result, (351 / 170). Understanding how to use such tools effectively is an essential skill in modern mathematics education.