Question

In Exercises \(35-38,\) write an equivalent series with the index of summation beginning at \(n=1\). $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{2 n+1} $$

Short answer

The equivalent series with the index of summation beginning at \(n=1\) is \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^{2n-1}}{2n-1}\).

Step-by-step solution

Identification of the pattern

Identify the overall pattern from the given series. The pattern here is \(\frac{(-1)^n x^{2n+1}}{2n+1}\) where the index \(n\) starts from 0.

Adjusting the index

Shift the index \(n\) from 0 to 1 by replacing the original \(n\) with \(n-1\) in the series. This is done to account for the fact that the series should now start from \(n=1\) instead of \(n=0\). So, our modified series becomes \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^{2(n-1)+1}}{2(n-1)+1}\).

Simplifying the series

Simplify the series obtained from Step 2. Distribute, where possible, inside the exponent and denominator. After simplifying, the equivalent series becomes: \(\sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^{2n-1}}{2n-1}\).

Key concepts

Infinite Series

An infinite series is a sum of an infinite sequence of terms; that is, it extends indefinitely. In the context of mathematics, an infinite series provides a framework for adding up the terms of a sequence to find their sum, even if there are infinitely many. A crucial point of studying an infinite series lies in understanding whether it converges or diverges.

Convergence of an infinite series occurs when the sum of the series approaches a finite value as more terms are added. Divergence, on the other hand, means the sum does not settle at a finite number, regardless of how many terms are included. The given exercise \( \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{2 n+1} \) presents an infinite series that can be analyzed for convergence using various tests, such as the alternating series test or the ratio test. Understanding convergence is essential because it assures us that the series has a well-defined value, making it meaningful and useful in applications across different fields of science and engineering.

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of terms. It is characterized by the Greek capital letter sigma (\(\Sigma\)). This notation is highly efficient, enabling mathematicians and students alike to express long and potentially complicated sums neatly and compactly.

The general form of summation notation includes an expression for the terms we want to add, an index that counts through the terms, a lower bound to indicate where to start the summation, and an upper bound to show where the summation ends. For example, the series \( \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2 n+1}}{2 n+1} \) uses summation notation to indicate that we are summing the terms \( \frac{(-1)^{n} x^{2 n+1}}{2 n+1} \) for each value of \(n\) starting from 0 and continuing indefinitely. It's essential for students to grasp this notation as it's not only a common tool for expressing series but also for calculating terms in sequences and operations on sets in higher-level mathematics.

Series Transformation

Series transformation refers to the process of modifying a series to achieve a particular form or property while maintaining the series' value or convergence behavior. Such transformations are common in calculus and analysis, where they are used for various purposes like simplification, comparison, or integration.

For instance, the transformation process in the step-by-step solution involves adjusting the index of summation. The original series starts with an index \(n=0\), but to meet the exercise requirement, we need to modify it to start at \(n=1\). This subtle shift, \(n\) to \(n-1\), preserves the series' integrity, meaning we still sum over the same terms, just changing the 'label' we use to count them. After altering the index, the series \( \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^{2n-1}}{2n-1} \) remains equivalent to the original, though it now starts at \(n=1\). Being adept at series transformation is a powerful skill as it often simplifies the work with series and helps in proving convergence or finding sums analytically.