A-Level Mathematics 9709 - Pure Mathematics 3 (P3)

Algebra: Partial Fractions, Series, Binomial Expansion

This section covers key algebraic techniques essential for success in Pure Mathematics 3. We will explore partial fractions, series expansion (including Taylor and Maclaurin series), and the binomial theorem.

1. Partial Fractions

Partial fractions are a method for decomposing a rational function into a sum of simpler fractions. This is particularly useful for integrating rational functions.

1.1 Indefinite Integral

When the degree of the numerator is greater than or equal to the degree of the denominator, we first perform polynomial division to obtain a remainder. The rational function can then be written as the sum of a polynomial and a proper fraction.

1.2 Simple and Repeated Factors

The denominator can be factored into simple linear factors (e.g., $x-a$) and repeated linear factors (e.g., $(x-a)^n$) and irreducible quadratic factors (e.g., $x^2 + bx + c$).

The partial fraction decomposition will have the form:

$$ \frac{P(x)}{Q(x)} = \frac{A}{x-a} + \frac{B}{(x-a)^n} + \frac{C}{x-r} + \frac{D}{(x-r)^m} + \dots $$

where A, B, C, and D are constants to be determined.

1.3 Determining the Coefficients

The coefficients A, B, C, and D are found by multiplying both sides of the equation by the original denominator $Q(x)$ and then equating coefficients of powers of $x$.

2. Series Expansion

A series expansion is an infinite sum of terms that represents a function. Two important types of series expansions are Taylor series and Maclaurin series.

2.1 Maclaurin Series

A Maclaurin series is a Taylor series centered at $x=0$. It is given by:

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \dots $$

where $f^{(n)}(0)$ denotes the nth derivative of $f$ evaluated at $x=0$ and $n!$ is the factorial of $n$.

2.2 Taylor Series

A Taylor series is a Taylor series centered at a point $a$. It is given by:

$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots $$

2.3 Series for Common Functions

Some common functions have known series expansions:

• $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \dots$
• $\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots$
• $\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots$ (for $|x| < 1$)
• $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots$ (for $|x| < 1$)
• $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots$

3. Binomial Expansion

The binomial theorem provides a formula for expanding $(a+b)^n$ for any real number $n$.

3.1 Formula

The binomial theorem states:

$$ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \dots + \binom{n}{n}b^n $$

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.

3.2 General Term

The general term in the binomial expansion is given by:

$$ T_{k+1} = \binom{n}{k} a^{n-k} b^k $$

3.3 Applications

The binomial theorem is used to calculate powers of binomials, particularly when $n$ is a non-negative integer. It is also used in probability and statistics.

Topic	Key Concepts
Partial Fractions	Decomposition of rational functions, simple and repeated factors, determining coefficients.
Series Expansion	Maclaurin series, Taylor series, series for common functions (sin, cos, 1/(1-x), ln(1+x), e^x).
Binomial Expansion	Binomial theorem, binomial coefficients, general term, applications.