Problem 9 - Advanced Engineering Mathematics

Question

Rewrite the given power series so that its general term involves \(x^{k}\). $$ \sum_{n=1}^{\infty} n c_{n} x^{n+2} $$

Step-by-Step Solution

Answer

The series is rewritten as \( \sum_{n=1}^{\infty} (n c_{n} x^2) x^n \).

Step 1: Identify the Series and the General Term

The given power series is \( \sum_{n=1}^{\infty} n c_{n} x^{n+2} \). The general term of this series is \( n c_{n} x^{n+2} \).

Step 2: Express the General Term with the Same Base

The term \( x^{n+2} \) can be rewritten as \( x^{n} \cdot x^{2} \) to separate it into a product involving \( x^{k} \), specifically \( x^{n} \).

Step 3: Update the General Term

Now, rewrite the general term as \( n c_{n} x^{n} x^{2} = n c_{n} x^2 \cdot x^n \).

Step 4: Rewrite the Entire Series

By substituting the expression from Step 3 back into the series, the power series becomes \( \sum_{n=1}^{\infty} (n c_{n} x^2) x^n \). Here, \( x^n \) is isolated, which makes it the general term expressed with \( x^k \).

Key Concepts

General Term

In the context of a power series, the **general term** is an essential building block. It represents the single term that recurs in the series, or more specifically, the expression that defines each element in the sequence. For a given power series, identifying the general term is crucial because it allows for the manipulation and transformation of the series.
In the exercise given, the power series is expressed as \( \sum_{n=1}^{\infty} n c_{n} x^{n+2} \). Here, the general term can be dissected into fundamental parts: the coefficient \( n c_{n} \), and the exponential component \( x^{n+2} \).
To align this term with the pattern of \( x^k \), it's useful to separate \( x^{n+2} \) into \( x^n \) and \( x^2 \). This manipulation helps in clearly representing the term with respect to \( x^n \), which simplifies the process of classic power series analysis and manipulation.

Infinite Series

Understanding an **infinite series** is key when working with power series. An infinite series consists of a series of numbers added together that do not terminate. In mathematics, particularly calculus, these series are vital for expressing functions as sums of infinite terms.
The given exercise presents a power series \( \sum_{n=1}^{\infty} n c_{n} x^{n+2} \), signifying that the series is infinite and commences with \( n=1 \). This means you are summing an endless number of terms without a natural endpoint, each defined by the general term.
Working with an infinite series requires understanding both the convergence of the series, if applicable, and the relationship between terms. This knowledge is fundamental for manipulating the series into a desired form, such as isolating particular powers of \( x \), as is done by restructuring terms to feature \( x^n \).
Mastery of infinite series concepts allows for a deeper comprehension of mathematical functions and their representations.

Algebraic Manipulation

**Algebraic manipulation** is a fundamental skill in dealing with power series and mathematical expressions in general. It involves transforming and reorganizing expressions in such a way to fulfill specific mathematical requirements or improve clarity.
In the exercise, algebraic manipulation was performed to transform the component \( x^{n+2} \) into \( x^n \cdot x^2 \). This kind of step is crucial as it helps in expressing the general term with the desired form \( x^k \).
Using algebraic rules and identities, such as the laws of exponents, facilitates such transformations. For instance, recognizing that \( a^{m+n} \) can be rewritten as \( a^m \cdot a^n \) is vital when separating terms.
This insight into manipulation serves to not only reorder expressions but also to pave the way for further simplifications or derivations. Such skills are invaluable in any advanced mathematical tasks, enabling complex expressions to become manageable and comprehensible.