Series and Sigma Notation with Binomial Expansion

Series, Sigma Notation and the Binomial Expansion

This topic extends your work on series to cover the binomial expansion for both positive integer and rational exponents, along with formal use of sigma notation and convergence tests.

The Binomial Expansion for Positive Integer n

For any positive integer n:

(a + b)ⁿ = Σ(r=0 to n) ⁿCᵣ · aⁿ⁻ʳ · bʳ

where ⁿCᵣ = n! / (r!(n-r)!) is the binomial coefficient (read "n choose r").

The expansion has n + 1 terms and the coefficients match the entries of Pascal's triangle.

Worked Example: Expand (2x + 3)⁴.

(2x + 3)⁴ = ⁴C₀(2x)⁴ + ⁴C₁(2x)³(3) + ⁴C₂(2x)²(3²) + ⁴C₃(2x)(3³) + ⁴C₄(3⁴)

= 16x⁴ + 4(8x³)(3) + 6(4x²)(9) + 4(2x)(27) + 81

= 16x⁴ + 96x³ + 216x² + 216x + 81

Finding a Specific Term

The (r + 1)th term of (a + b)ⁿ is ⁿCᵣ · aⁿ⁻ʳ · bʳ. This is useful when you need one particular coefficient without expanding everything.

Worked Example: Find the coefficient of x³ in (1 + 2x)⁸.

The general term is ⁸Cᵣ · 1⁸⁻ʳ · (2x)ʳ = ⁸Cᵣ · 2ʳ · xʳ

For x³: r = 3. Coefficient = ⁸C₃ × 2³ = 56 × 8 = 448

The Binomial Expansion for Rational n

When n is not a positive integer (for example n = -1, 1/2, -3), the expansion of (1 + x)ⁿ becomes an infinite series:

(1 + x)ⁿ = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ...

This series is valid only when |x| < 1 (the series converges).

Important: The expansion must start with (1 + ...)ⁿ. If you have (a + bx)ⁿ, factor out aⁿ first:

(a + bx)ⁿ = aⁿ(1 + bx/a)ⁿ, valid when |bx/a| < 1, i.e. |x| < a/b.

Worked Example: Expand (1 + x)⁻² up to the term in x³. State the values of x for which the expansion is valid.

(1 + x)⁻² = 1 + (-2)x + (-2)(-3)x²/2! + (-2)(-3)(-4)x³/3! + ...

= 1 - 2x + 3x² - 4x³ + ...

Valid for |x| < 1.

Worked Example: Find the first three terms of (4 - x)^(1/2).

Factor out 4^(1/2) = 2:

2(1 - x/4)^(1/2) = 2[1 + (1/2)(-x/4) + (1/2)(-1/2)(-x/4)²/2! + ...]

= 2[1 - x/8 + (-1/4)(x²/16)/2 + ...]

= 2[1 - x/8 - x²/128 + ...]

= 2 - x/4 - x²/64 + ...

Valid for |x/4| < 1, i.e. |x| < 4.

Using Partial Fractions with Binomial Expansion

A powerful technique: decompose a rational function into partial fractions, then expand each fraction using the binomial theorem.

Worked Example: Express (3 + x)/((1 + x)(1 - 2x)) in partial fractions and expand up to x².

Partial fractions: 2/(1 + x) + 1/(1 - 2x) (verify by recombining).

2(1 + x)⁻¹ = 2(1 - x + x² - ...) = 2 - 2x + 2x²

1(1 - 2x)⁻¹ = 1 + 2x + 4x² + ...

Adding: 3 + 0x + 6x² + ... = 3 + 6x²

Valid for |x| < 1 and |2x| < 1, so |x| < 1/2 (the tighter restriction).

Approximations Using the Binomial Expansion

You can use the expansion to approximate values by substituting a suitable x.

Example: Estimate √(1.02) using (1 + x)^(1/2) with x = 0.02.

√(1.02) ≈ 1 + (1/2)(0.02) + (1/2)(-1/2)(0.02)²/2 = 1 + 0.01 - 0.00005 = 1.00995

(Calculator gives 1.009950494... — excellent accuracy with just three terms.)

Exam Tips

• For the integer binomial expansion, always use ⁿCᵣ — do not expand by multiplying out repeatedly.
• For rational/negative n, the expansion is infinite and you must state the validity condition |x| < 1.
• When factoring out to create (1 + ...)ⁿ form, be careful with signs — (2 - 3x)⁻¹ = (1/2)(1 - 3x/2)⁻¹.
• Read the question carefully: "up to and including the term in x³" means four terms (constant, x, x², x³).
• In approximation questions, state what value of x you are substituting and show that it lies within the valid range.