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Chapter 9: Problem 48

Writing an Equivalent Series In Exercises \(45-48,\) write an equivalent series with the index of summation beginning at \(n=1 .\) $$\sum_{n=2}^{\infty} \frac{x^{3 n-1}}{(2 n-1) !}$$

Short Answer

Expert verified
The requested series with the index of summation beginning at \(n=1\) is given by \(\sum_{m=1}^{\infty} \frac{x^{3m+2}}{(2m+1)!}\).

Step by step solution

01

Note the Original Series and the Desired Change

The given infinite series is \(\sum_{n=2}^{\infty} \frac{x^{3n-1}}{(2n-1)!}\). The desired series needs to start with \(n=1\), instead of \(n=2\). Therefore, a change in the index of summation will occur.
02

Introduce new variable for the index of summation

Let's introduce a new variable \(m\) where \(m = n - 1\). Thus, when \(n=2\), \(m=1\). This allows the index of summation to start at 1.
03

Make substitutions in the original series

Now substitute \(m = n - 1\) into the original series which is expressed as \\(\sum_{n=2}^{\infty} \frac{x^{3n-1}}{(2n-1)!}\). This results in the series \(\sum_{m=1}^{\infty} \frac{x^{3(m+1)-1}}{(2(m+1)-1)!}\).
04

Simplify the new series

The series can further be simplified to: \(\sum_{m=1}^{\infty} \frac{x^{3m+2}}{(2m+1)!}\). This is the equivalent series starting at \(n = 1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Index of Summation
The index of summation is akin to a cursor moving through a list, pointing to each term that needs to be included in a series. When dealing with series in mathematics, especially infinite series, the index of summation tells us where to start and where to end.
In our case, we encountered a series where the terms started from an index of 2. Converting it to start from 1 required a transformation of the variable. This is more than a trivial step, as it ensures the terms of the series align with the new index. By substituting with a new variable, we achieve this realignment without altering the series' value. Changing the index is a powerful tool and must be done with consideration to maintain the integrity of the sequence.
Infinite Series
An infinite series is a sum of infinitely many terms, extending without end. It's a concept that stupefies the imagination 鈥� how can we possibly add up an infinite number of things? Yet, mathematically, such series can converge to finite values, telling a profound story about the nature of infinity.
The series we explored was just such an infinite series, represented by a sum that continues indefinitely. The fascinating part is when we can meaningfully talk about the sum of an infinite series, even assigning it a precise value. Ancillary concepts like convergence tests help elucidate when an infinite series can be summed, but that's a topic for another day.
Factorial Notation
Factorial notation is a shorthand to represent the product of an integer and all the positive integers below it, down to 1. It's denoted by an exclamation mark (!). For instance, 5! is equal to 5 脳 4 脳 3 脳 2 脳 1.
In our series, the factorial appears in the denominator and it has crucial implications for the series' convergence. Factorials grow extremely quickly, much faster than exponential expressions grow, which influences how the terms of the series behave as the index increases. This a key idea when considering the convergence of series with factorial terms.
Series Simplification
Series simplification is the process of methodically reducing a series to a more manageable or recognizable form. This often includes altering the series index, combining like terms, or applying mathematical identities that transform the series without changing its value.
During our series manipulation, simplification was the final stage. Unlike algebraic simplification, simplifying a series involves ensuring that the terms are as straightforward as possible. This can reveal patterns, facilitate comparison with known series, or clarify the properties of the series we're examining, like its convergence. It's an elemental part of working with series and can offer profound insights into their nature.

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Most popular questions from this chapter

Maclaurin Series Explain how to use the power eries for \(f(x)=\arctan x\) to find the Maclaurin series for \(g(x)=\frac{1}{1+x^{2}}\) What is another way to find the Maclaurin series for \(g\) using a power series for an elementary function?

Writing an Equivalent Series In Exercises \(45-48,\) write an equivalent series with the index of summation beginning at \(n=1 .\) $$\sum_{n=0}^{\infty}(-1)^{n+1}(n+1) x^{n}$$

$$\begin{array}{l}{\text { Investigation The interval of convergence of the geometric }} \\ {\text { series } \sum_{n=0}^{\infty}\left(\frac{x}{4}\right)^{n} \text { is }(-4,4)}\\\ {\text { (a) Find the sum of the series when } x=\frac{3}{2} . \text { Use a graphing }} \\ {\text { utility to graph the first six terms of the sequence of partial }} \\ {\text { sums and the horizontal line representing the sum of the }} \\ {\text { series. }} \\ {\text { (b) Repeat part (a) for } x=-\frac{5}{2} \text { . }}\\\ {\text { (c) Write a short paragraph comparing the rates of convergence }} \\ {\text { of the partial sums with the sums of the series in parts (a) }} \\ {\text { and (b). How do the plots of the partial sums differ as they }} \\ {\text { converge toward the sum of the series? }} \\ {\text { (d) Given any positive real number } M, \text { there exists a positive }} \\\ {\text { integer } N \text { such that the partial sum }}\end{array}$$ $$\sum_{n=0}^{N}\left(\frac{5}{4}\right)^{n}>M$$

Finding the Interval of Convergence In Exercises \(15-38\) , find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$\sum_{n=1}^{\infty} \frac{n}{n+1}(-2 x)^{n-1}$$

Approximating an Integral In Exercises \(63-70\) , use a power series to approximate the value of the definite integral with an error of less than \(0.0001 .\) (In Exercises 65 and \(67,\) assume that the integrand is defined as 1 when \(x=0 . )\) $$\int_{0}^{1} e^{-x^{3}} d x$$
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