Question

Rewrite the given expression as a sum whose generic term involves $x^n.$ $$x\sum _{n=2}^{\mathrm{\infty }}n(n-1)a_n{x}^{n-2}+\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$$

Short answer

Question: Express the following as a sum whose generic term involves $x^n$: $$x\sum _{n=2}^{\mathrm{\infty }}n(n-1)a_n{x}^{n-2}+\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$$ Answer: $$S(x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n[(n-1)n{x}^n+x^n]$$

Step-by-step solution

Observe the given expressions

Notice that the first sum has terms with exponent $x^{n-2}$ and coefficients $n(n-1)a_n$, and the second sum has exponent $x^n$ and coefficients $a_n$. Our goal is to combine these expressions into a single power series.

Combine the expressions

We need to ensure that the exponents of both summations match so that we can combine these series. In the first summation, we substitute $(n-2)$ with $m$ (where $m=n-2$). Then $n=m+2$. Rewrite the first summation with the appropriate substitutions: $$x\sum _{n=2}^{\mathrm{\infty }}n(n-1)a_n{x}^{n-2}=x\sum _{m=0}^{\mathrm{\infty }}(m+2)(m+1)a_{m+2}{x}^m$$ Now both summations are in terms of $x^m$.

Combine both sums

Rewrite the expression combining both sums: $$S(x)=x\sum _{m=0}^{\mathrm{\infty }}(m+2)(m+1)a_{m+2}{x}^m+\sum _{m=0}^{\mathrm{\infty }}{a}_m{x}^m$$

Factor out the common $x^m$ term

Factor out the common $x^m$ term and rewrite the expression as one summation: $$S(x)=\sum _{m=0}^{\mathrm{\infty }}[(m+2)(m+1)a_{m+2}{x}^{m+1}+a_m{x}^m]$$

Rewrite as a generic term $x^n$

Since the exponent $m+1$ in the first part of the summation ranges from $1$ to $\mathrm{\infty }$, we can rewrite the series with $n$ as the exponent: $$S(x)=\sum _{n=0}^{\mathrm{\infty }}[(n-1)n{a}_n{x}^n+a_n{x}^n]$$

Group the coefficients of $x^n$

To achieve the final form, we need to group the coefficients of $x^n$. Combine terms with $a_n$: $$S(x)=\sum _{n=0}^{\mathrm{\infty }}{a}_n[(n-1)n{x}^n+x^n]$$ Therefore, the expression has been rewritten as a sum whose generic term involves $x^n$.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. In essence, they are used to model real-world systems where change is involved. For instance, if you think about the growth of a population, how a car slows down, or how a heat spreads through a room, these are all phenomena that can be described using differential equations.

There are many types of differential equations, such as:

• Ordinary Differential Equations (ODEs) - involve functions of a single variable and their derivatives.
• Partial Differential Equations (PDEs) - involve functions of multiple variables and their derivatives.

In the context of power series, differential equations can sometimes be solved by expressing the solution as a series. This involves finding coefficients that satisfy the equation when summed. Using series to solve differential equations is particularly useful when the solution cannot be easily expressed using standard functions.

Mathematical Analysis

Mathematical analysis is a branch of mathematics that deals with understanding the properties of real and complex numbers, functions, and sequences. It involves analyzing how quantities change and how to understand convergence, continuity, and limits. Analysis is crucial to calculus and it provides the foundation for understanding mathematical rigor and formal proofs.

Key aspects include:

• Limits - investigating how functions behave as variables approach a certain point.
• Continuity - ensuring that small changes in input result in small changes in output.
• Convergence - studying if a sequence or series approaches a certain value as terms continue.

When working with power series, mathematical analysis allows us to ensure the series converges and represents a function accurately within a radius of convergence.

Series Expansion

Series expansion helps express functions in terms of a sum of a sequence of terms. This is extremely powerful in mathematics because it allows complex functions to be expressed in simpler terms, making computation and calculus on these functions more manageable.

A power series is a specific type of series expansion given by:$$a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots =\sum _{n=0}^{\mathrm{\infty }}{a}_n{x}^n$$This represents a function as a sum of its terms. The process of series expansion can often involve manipulating the indices within sums, matching terms to find a general solution format, like we did in our exercise.

Series expansions are invaluable tools for approximating functions, especially in cases where exact forms are hard to obtain or integrate. They also facilitate the analysis of functions, allowing us to see patterns and relationships that are not immediately obvious.