Question

Given that $$ \frac{1}{1-x}=\sum_{n=0}^{\infty} x^{n}, \quad|x|<1 $$ use Theorem 4.5 .4 to express \(\sum_{n=0}^{\infty} n^{2} x^{n}\) in closed form.

Short answer

Answer: The closed-form expression of the power series \(\sum_{n=0}^{\infty} n^2 x^n\) is \(\sum_{n=0}^{\infty} (n+1) x^{n+1}\).

Step-by-step solution

Differentiate the given function twice

We will first differentiate \(\frac{1}{1-x}\) with respect to x. $$ \frac{d}{dx}\left(\frac{1}{1-x}\right) = \frac{d}{dx}(1-x)^{-1} = (-1)(1-x)^{-2}(-1) = \frac{1}{(1-x)^{2}} $$ Next, we will differentiate the result with respect to x again. $$ \frac{d^2}{dx^2}\left(\frac{1}{1-x}\right) = \frac{d}{dx}\left(\frac{1}{(1-x)^{2}}\right) = 2(1-x)^{-3}(-1)=\frac{2}{(1-x)^{3}} $$

Apply Theorem 4.5.4

According to Theorem 4.5.4, if the function \(\frac{1}{1-x}\) has a power series representation \(\sum_{n=0}^{\infty} x^n\) for \(|x|<1\), then its second derivative also has a power series representation. The second derivative we found is \(\frac{2}{(1-x)^3}\). Now we need to find the power series representation of this function.

Multiply the power series representation of the first derivative by x

Let's first find the power series representation of the first derivative \(\frac{1}{(1-x)^{2}}\). Start with the power series representation of the given function, \(\sum_{n=0}^{\infty} x^n\), and multiply it by x. $$ x\sum_{n=0}^{\infty} x^n = \sum_{n=1}^{\infty} x^n $$

Differentiate the result and apply Theorem 4.5.4

Now differentiate the result with respect to x. $$ \frac{d}{dx}\left(\sum_{n=1}^{\infty} x^n\right) = \sum_{n=1}^{\infty} n x^{n-1} = \sum_{n=0}^{\infty} (n+1) x^{n} $$ This is the power series representation of the first derivative of \(\frac{1}{1-x}\), which is \(\frac{1}{(1-x)^{2}}\). Apply Theorem 4.5.4 again to find the power series representation of the second derivative, \(\frac{2}{(1-x)^3}\).

Find the closed-form expression of the power series

Now to express the power series \(\sum_{n=0}^{\infty} n^2 x^n\) in closed form, we just need to multiply the power series representation of the second derivative by x: $$ x\sum_{n=0}^{\infty} (n+1) x^{n} = \sum_{n=0}^{\infty} (n+1) x^{n+1} = \sum_{n=0}^{\infty} n^2 x^n $$ This is the closed-form expression of the power series we were looking for: $$ \sum_{n=0}^{\infty} n^2 x^n = \sum_{n=0}^{\infty} (n+1) x^{n+1} $$

Key concepts

Real Analysis

Real Analysis is a branch of mathematical analysis that deals with real numbers and real-valued functions. At its core, it involves the study of sequences and series, particularly with respect to their convergence properties. Power series, such as the one featured in the exercise \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^{n}, \quad|x|<1 \), are a prime example of such objects of study in real analysis. Understanding the conditions under which a power series converges is crucial. The given power series converges absolutely for \( |x| < 1 \), which is known as the series' radius of convergence. This specific interval allows for the manipulation and differentiation of the series within its radius of convergence, maintaining the integrity of the series' identity and providing valuable methods for deriving closed-form expressions like in the exercise.

Closed-Form Expression

A closed-form expression is an explicit equation that allows one to directly compute the value of a function or sequence without the need for iterative steps or summation. Closed-form expressions are highly sought in mathematics as they provide a succinct way to represent complex operations or infinite series. For instance, in our problem, we have a series \( \sum_{n=0}^{\infty} n^{2} x^{n} \), which does not initially appear to have a closed-form. However, by applying real analysis techniques such as series differentiation and integrating a corresponding simpler series, we transform it into a more manageable expression. Finding the closed-form equivalent then simplifies computation and aids in further mathematical exploration of the series in question. The process employed in the exercise represents a powerful strategy for converting many infinite series into closed-form expressions.

Series Differentiation

Series differentiation is a technique used to find the derivative of a power series term-by-term. It is a valuable tool because it allows the application of standard calculus techniques to infinite series. In real analysis, it is essential to verify that differentiating a power series within its interval of convergence does not alter the convergence. The exercise demonstrates an application of series differentiation, where the power series for \( \frac{1}{1-x} \) is differentiated twice, yielding a new series that represents the power series of \( \frac{2}{(1-x)^{3}} \) after appropriate manipulation. When students encounter a power series and are asked to find a closed-form expression for a more complex series (such as \( \sum_{n=0}^{\infty} n^{2} x^{n} \) in the exercise), series differentiation is a key step. The concept of series differentiation leverages the linearity of differentiation and the structure of power series to extend differentiation operations from polynomials to infinite series, a critical transition in higher mathematics.