Chapter 5: Problem 21
Rewrite the given expression as a sum whose generic term involves \(x^{n} .\) $$ \sum_{n=2}^{\infty} n(n-1) a_{n} x^{n-2} $$
Short Answer
Expert verified
Question: Rewrite the expression \(\sum_{n=2}^{\infty} n(n-1) a_{n} x^{n-2}\) as a sum with a generic term that involves \(x^{n}\).
Answer: The given expression can be rewritten as \(\sum_{m=0}^{\infty} (m+2)(m+1) a_{m+2} x^{m}\).
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Summation Notation
Summation notation, also known as sigma notation, is a convenient and compact way to write long sums. When dealing with a summation, the symbol \(\sum\) is used to represent the sum of a sequence of terms. The variable right below this symbol is the index of summation, which in our exercise is \(n\), and it changes from a lower bound to an upper bound. In the original expression \(\sum_{n=2}^{\infty} n(n-1) a_{n} x^{n-2}\), the index of summation \(n\) starts at 2 and goes to infinity, indicating that this is an infinite series.
Each term in the sequence is defined as a function of \(n\), which in this case is \(n(n-1) a_{n} x^{n-2}\). Summation notation helps in simplifying and understanding the structure of series, especially when working with complex terms or a large number of terms.
Each term in the sequence is defined as a function of \(n\), which in this case is \(n(n-1) a_{n} x^{n-2}\). Summation notation helps in simplifying and understanding the structure of series, especially when working with complex terms or a large number of terms.
Power Series
A power series is a type of infinite series that can be thought of as a polynomial with infinitely many terms, where each term involves a power of the variable \(x\). It is usually expressed in the form \(\sum_{n=0}^{\infty} a_n x^n\). Power series are used in various branches of mathematics to represent functions and to solve differential equations.
The main advantage of power series is that they can converge to a wide range of functions within their interval of convergence and can be easily manipulated through differentiation and integration. In the exercise, we are dealing with a series that we'd like to express in terms of \(x^n\), leading us to adjust the series to a power series format.
The main advantage of power series is that they can converge to a wide range of functions within their interval of convergence and can be easily manipulated through differentiation and integration. In the exercise, we are dealing with a series that we'd like to express in terms of \(x^n\), leading us to adjust the series to a power series format.
Change of Index in Series
Changing the index of a series refers to the process where we replace the existing index of summation with a new variable. This technique is useful when the goal is to manipulate the series into a more familiar or useful form. In the step-by-step solution, the index \(n\) is replaced with \(m+2\), effectively shifting the entire series.
As a result, \(m = n - 2\), and when \(n = 2\), \(m\) will be 0, thereby changing the series so that it starts at \(m = 0\) instead of \(n = 2\). This re-indexing is particularly helpful in this context because it aligns the powers of \(x\) with the coefficients \(a_{m+2}\), simplifying the analysis and calculation of the series.
As a result, \(m = n - 2\), and when \(n = 2\), \(m\) will be 0, thereby changing the series so that it starts at \(m = 0\) instead of \(n = 2\). This re-indexing is particularly helpful in this context because it aligns the powers of \(x\) with the coefficients \(a_{m+2}\), simplifying the analysis and calculation of the series.
Infinite Series
An infinite series is the sum of the terms of an infinite sequence. When we talk about an infinite series, such as \(\sum_{n=2}^{\infty} n(n-1) a_{n} x^{n-2}\) from our exercise, it means that the series has an endless number of terms.
Convergence is a critical concept when dealing with infinite series, which means that as more and more terms are added, the sum approaches a certain finite limit. However, some infinite series do not converge to a limit; these are called divergent. The transformed series in our problem is also an infinite series, now starting with \(m = 0\), and it is representative of the original problem in a way that emphasizes each term involving a power of \(x\) as \(x^m\). It's important for students to understand the behavior and nature of infinite series within the context of convergence and divergence.
Convergence is a critical concept when dealing with infinite series, which means that as more and more terms are added, the sum approaches a certain finite limit. However, some infinite series do not converge to a limit; these are called divergent. The transformed series in our problem is also an infinite series, now starting with \(m = 0\), and it is representative of the original problem in a way that emphasizes each term involving a power of \(x\) as \(x^m\). It's important for students to understand the behavior and nature of infinite series within the context of convergence and divergence.
