Problem 109 - Precalculus with Limits

Question

In Exercises 103-112, use sigma notation to write the sum. \( \dfrac{1}{1^2} - \dfrac{1}{2^2} + \dfrac{1}{3^2} - \dfrac{1}{4^2} + \cdots - \dfrac{1}{20^2} \)

Step-by-Step Solution

Answer

The sigma notation for \( \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \cdots - \frac{1}{20^2} \) is \( \sum_{n=1}^{20} (-1)^{n+1} \frac{1}{n^2} \)

Step 1: Identify the series and pattern

The given series is an alternating series since it's pattern alternates between positive and negative, the series is \( \frac{1}{n^2} \), where \( n \) starts at 1 and upto 20 and alternating in sign. Thus, the series is: \( \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \cdots - \frac{1}{20^2} \). The pattern of the series suggests that elements are positive when the term is odd and negative when the term is even.

Step 2: Formation of generalized series

The element of the series can be generalized to \( (-1)^{n+1} \frac{1}{n^2} \) for \( n = 1 \) to \( n = 20 \). This is due to the fact that for odd n, \( (-1)^{n+1} = 1 \) rendering the term positive, and for even n, \( (-1)^{n+1} = -1 \), making the term negative, which is consistent with the given pattern

Step 3: Apply sigma notation

Now that a general term for the series has been established, it can be represented using sigma notation. Therefore, the sum \( \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \cdots - \frac{1}{20^2} \) can be written in sigma notation as \( \sum_{n=1}^{20} (-1)^{n+1} \frac{1}{n^2} \)

Key Concepts

Alternating Series

When approaching sequences and series, one may encounter the term 'alternating series.' An alternating series is characterized by consecutive terms having opposite signs. For example, each successive term may switch from positive to negative or vice versa. This creates a pattern in which the signs of the terms alternate.

Understanding alternating series is crucial as they appear frequently in mathematics, particularly in calculus, where they are used for tests of convergence, power series, and more. In the example provided, we see such a series, where the terms alternate as positive and negative in a consistent pattern: positive for odd terms and negative for even terms. A better understanding of such series often involves unraveling this pattern to write the series in a more standardized form such as sigma notation.

General Term of a Series

The 'general term' of a series essentially captures the rule or formula that defines every term in the series based on its position. Recognizing this rule is a fundamental step in working with series, as it allows us to not only describe the series in concise mathematical language but also manipulate it for further analysis.

For instance, in the alternating series example, the general term is identified as \((-1)^{n+1} \frac{1}{n^2}\), where \(n\) is the position of a term in the sequence. This incorporates both the alternating sign and the square in the denominator. By determining the general term, we can easily describe and work with all terms of the series, regardless of its length.

Sum of Series

The 'sum of a series' is the result of adding all the terms of the series together. Sigma notation is particularly helpful for representing these sums, especially when dealing with a series that has many terms or follows a complex pattern. Once the general term of a series is known, the sigma notation provides a way to express the entire sum compactly.

To write a sum in sigma notation, identify the general term and then decide the bounds (starting and ending values) for the index of summation. In the example provided, the sum includes terms from \(n=1\) to \(n=20\), making it a finite sum. The key to finding the sum of a series lies in understanding both the general term and the range over which it applies.

Precalculus

Precalculus is a mathematical course that prepares students for the study of calculus. It encompasses a wide array of topics including functions, complex numbers, vectors, matrices, and an introduction to sequences and series. When dealing with series in precalculus, it is vital to understand the concept of sigma notation, general terms, and how to find the sum of series to cultivate a strong foundation for calculus concepts.

Manipulating series and understanding their behaviors in precalculus can significantly help students as they progress into more advanced studies, where series play a pivotal role in topics like series convergence, Taylor and Fourier series, as well as applications in physics, economics, and other sciences.