91Ó°ÊÓ

The process of decomposing a rational function into partial fractions is an essential algebraic technique. In this context, a rational function is an expression of the form \(\frac{P(x)}{Q(x)}\), where both \(P(x)\) and \(Q(x)\) are polynomials. Our goal is to express this rational function as a sum of simpler fractions, which makes it easier to manipulate, especially when integrating or transforming into series.In the exercise, the denominator \(x^2 + 3x + 2\) was factored into \((x+1)(x+2)\). This factorization allows us to express the function as:

• \(\frac{A}{x+1} + \frac{B}{x+2}\)

By setting up the equation \(2x + 3 = A(x+2) + B(x+1)\) and equating the coefficients, we found the values of \(A\) and \(B\). Solving gives \(A = 1\) and \(B = 1\). Therefore, the decomposed form of the function is \(\frac{1}{x+1} + \frac{1}{x+2}\).This step is crucial because it simplifies the process of finding a power series, making the function easier to handle in subsequent calculations.

When a power series is formed, determining its interval of convergence is essential. The interval shows the range of \(x\) values for which the series converges to a finite sum.For the series \(\sum_{n=0}^{\infty} (-x)^n\), it is a geometric series. A geometric series converges when its common ratio's absolute value is less than one. Here, the common ratio is \(-x\), and thus it converges if \(|x| < 1\).Similarly, the series \(\sum_{n=0}^{\infty} \left(-\frac{x}{2}\right)^n\) also has convergence determined by the common ratio \(-\frac{x}{2}\). This series converges when \(|x/2| < 1\) or equivalently \(|x| < 2\).The overall power series convergence depends on the intersection of these intervals. Here, both series must converge, limiting our interval to \(|x| < 1\). Therefore, the interval of convergence is from \(-1 < x < 1\). This indicates that within this open interval, the power series represents the function accurately.

A geometric series is one of the simplest types of series, defined as a series where each term is a constant multiple of the previous one. The general form of a geometric series is:

• \(a + ar + ar^2 + ar^3 + \ldots\)

where \(a\) is the first term and \(r\) is the common ratio.For convergence, the absolute value of the common ratio \(r\) must be less than 1 (\(|r| < 1\)). When it converges, the sum of the infinite geometric series is given by the formula:

• \(\frac{a}{1 - r}\)

In the exercise, each fraction of the decomposed function was expressed as a geometric series:

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

These expressions were crucial for forming the power series of the original function, taking advantage of the well-known properties of geometric series.