Problem 21 - Calculus: Early Transcendentals

Question

Find a power series representation for \( f, \) and graph \( f \) and several partial sums \( s_n(x) \) on the same screen. What happens as \( n \) increases? \( f(x) = \frac {x^2}{x^2 + 1} \)

Step-by-Step Solution

Answer

As \( n \) increases, the partial sums of the series converge to \( f(x) = \frac{x^2}{x^2+1} \).

Step 1: Express the function in a form suitable for a power series

First, observe that the function can be rewritten as: \[ f(x) = x^2 \cdot \frac{1}{1 - (-x^2)} \]. This expression suggests a geometric series with the form \( \frac{1}{1-r} = \sum_{n=0}^{\infty} r^n \) where \( r = -x^2 \).

Step 2: Write the power series for the modified function

Using the geometric series formula, the term \( \frac{1}{1 + x^2} \) can be expanded as: \[ \sum_{n=0}^{\infty} (-1)^n x^{2n} \].

Step 3: Combine the power series with the multiplication factor

Since \( f(x) = x^2 \cdot \frac{1}{1 - (-x^2)} \), multiply the power series term by \( x^2 \): \[ f(x) = x^2 \cdot \sum_{n=0}^{\infty} (-1)^n x^{2n} = \sum_{n=0}^{\infty} (-1)^n x^{2n+2} \].

Step 4: Identify the general formula for the series

The power series representation for the function is: \[ \sum_{n=0}^{\infty} (-1)^n x^{2n+2} \]. This notation captures the essence of the function as an infinite series.

Step 5: Graph the function and partial sums

To visualize, graph \( f(x) = \frac{x^2}{x^2 + 1} \) along with partial sums such as: \[ s_n(x) = \sum_{k=0}^{n} (-1)^k x^{2k+2} \] for increasing \( n \). As \( n \) increases, the partial sums appear to converge to the actual function.

Key Concepts

Geometric Series

A geometric series is a sum of the form

• \( a + ar + ar^2 + ar^3 + \ldots \)

Here, each term is obtained by multiplying the previous term by a constant, known as the ratio. It can be easily represented using the formula \( \frac{1}{1-r} = \sum_{n=0}^{\infty} r^n \), where \( r \) is the common ratio.

In the context of the given exercise, we have the expression \( \frac{1}{1 - (-x^2)} \), which means the common ratio is \( -x^2 \).

This allows us to rewrite the function as a geometric series:

• \( \sum_{n=0}^{\infty} (-1)^n x^{2n} \)

The negative sign accounts for the negative ratio, while the power of \( x \) adjusts for the exponent in the series.

Using geometric series is fundamental in finding power series representations, as it provides a straightforward method to express functions as sums of infinite terms.

Partial Sums

Partial sums are a key concept when working with power series. They allow us to approximate the value of a series by summing a finite number of terms.

For a series \( \sum_{n=0}^{\infty} a_n \), a partial sum \( s_n \) is defined as

• \( s_n = a_0 + a_1 + a_2 + \ldots + a_n \)

In this exercise, partial sums take the form

• \( s_n(x) = \sum_{k=0}^{n} (-1)^k x^{2k+2} \)

Here, each term \( (-1)^k x^{2k+2} \) corresponds to adding up to a finite \( n \), which gives us an approximation of the infinite series.

Graphing these partial sums alongside the actual function \( f(x) \) helps us see how closely they match as \( n \) increases. With each additional term, the approximation becomes more accurate, illustrating the convergence of the series to the function.

Function Representation

Representing functions as power series can simplify complex algebraic expressions into more manageable infinite series.

In this exercise, the function \( f(x) = \frac{x^2}{x^2 + 1} \) was represented as

• \( \sum_{n=0}^{\infty} (-1)^n x^{2n+2} \)

This power series is obtained by employing the geometric series formula and manipulating the original expression.

The beauty of power series lay in their adaptability. They enable us to work with functions analytically over an interval where the series converges, providing insights that discrete terms might not reveal.

Additionally, graphing this representation and its partial sums showcases how continuous and smooth functions can be expressed through a series of simpler polynomials. This is especially valuable in applications like calculus and physics, where understanding the behavior of functions is essential.