Problem 37 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Analyzing Infinite Series

The given series is \( \sum_{n=0}^{\infty} \frac{x^{2 n+1}}{(2 n+1) !} \). The index of summation in this series starts at \( n=0 \). The goal is to make an equivalent series with the index of summation starting at \( n=1 \). To achieve this, we need to substitute \( n-1 \) for \( n \) in the expression of the series, and adjust the limits of the sum.

Step 2: Substitute \( n-1 \) for \( n \) and Adjust the Limits

Replace every \( n \) in the expression by \( n-1 \), we get \( x^{2*(n-1)+1} \)/\((2*(n-1)+1)!\). The index after substitution will now start from \( n=1 \) instead of \( n=0 \), so the series now becomes: \( \sum_{n=1}^{\infty} \frac{x^{2 (n-1) +1}}{(2(n-1) +1)!} \).

Step 3: Simplify the New Series Expression

The new series expression \( \sum_{n=1}^{\infty} \frac{x^{2 (n-1) +1}}{(2(n-1) +1)!} \) can be simplified to \( \sum_{n=1}^{\infty} \frac{x^{2n-1}}{(2n -1)!} \).

Key Concepts

Index of Summation

Understanding the index of summation is crucial when dealing with series in calculus. The index of summation, typically denoted as 'n' in mathematical notation, signifies the starting point and the sequence of terms being added in a series. In the given exercise, the series begins with an index of summation at n=0. However, for various reasons, including simplification or comparison, we might want to redefine the series to start from a different index, such as n=1.

To do this, we employ a substitution method, replacing every instance of n with (n-1). This alters the terms of the series, but not its value. The new series is equivalent to the original, but now with an index that starts at 1. This kind of manipulation is an essential skill for students tackling problems involving infinite series, as it helps in understanding the properties and behaviors of the series across different indexes.

Series Convergence

Determining the convergence of a series is a foundational aspect of working with infinite series. When we mention series convergence in calculus, we're talking about whether the terms of the series approach a specific value or if they spread out towards infinity. In the problem provided, the series is an infinite series, which means that it adds up an infinite number of terms.

However, not all infinite series converge. To check for convergence, certain tests can be applied, like the Ratio Test or the Root Test. If a series converges, then we can talk about its sum as a finite value, even though it's composed of infinitely many terms. Understanding whether a series converges and finding its sum if it does are both vital components of calculus, especially in the context of power series and their applications in functions approximation.

Factorial Notation

Factorial notation plays a significant role in series, especially when we deal with the terms of a series that involve factorial operations. The notation 'n!' (read as 'n factorial') means the product of all positive integers from 1 to n. For example, 4! equals 1 x 2 x 3 x 4, which is 24. Factorials often appear in the denominators of series expressions, especially those akin to the Taylor and Maclaurin series.

In the solved problem, the factorial appears within the series term \( (2n+1)! \). As the index of summation changes, so does the factorial expression. It is essential for students to become comfortable with this notation as calculating and manipulating factorials is a common requirement in many areas of advanced mathematics, from series expansion problems like this one to combinations and probabilities in statistics.