Sample Notes: Series

12. Series

12.1 Binomial Theorem for Expansion of (a + b)ⁿ

• The binomial theorem allows us to expand expressions of the form (a + b)ⁿ where n is a positive integer.
• The general formula is:

  (a + b)ⁿ = Σⁿᵣ₌₀ ᶜ(n, r) × aⁿ⁻ʳ × bʳ

  where ᶜ(n, r) (also written as nCr) is the binomial coefficient, defined as:

  ᶜ(n, r) = n! / (r!(n − r)!)

Example:

Expand (x + 2)³

→ Using binomial theorem:

(x + 2)³ = ᶜ(3, 0)x³(2)⁰ + ᶜ(3, 1)x²(2)¹ + ᶜ(3, 2)x¹(2)² + ᶜ(3, 3)x⁰(2)³
= 1×x³ + 3×x²×2 + 3×x×4 + 1×8
= x³ + 6x² + 12x + 8

12.2 General Term in a Binomial Expansion

• The general term in the expansion of (a + b)ⁿ is:

  Tᵣ₊₁ = ᶜ(n, r) × aⁿ⁻ʳ × bʳ

  where r = 0, 1, 2, …, n

Example:

Find the 4th term in the expansion of (2x − 1)⁵

→ Use the formula:
T₄ = ᶜ(5, 3) × (2x)² × (−1)³ = 10 × 4x² × (−1) = −40x²

12.3 Arithmetic and Geometric Progressions

Arithmetic Progression (A.P.)

• A sequence in which each term increases or decreases by a constant value (common difference, d).
• General term:
  uₙ = a + (n − 1)d
• Sum of first n terms:
  Sₙ = n/2 × [2a + (n − 1)d]

Example:

Find the 10th term and sum of first 10 terms of the sequence: 3, 7, 11, …

• a = 3, d = 4
• u₁₀ = 3 + 9×4 = 39
• S₁₀ = 10/2 × [2×3 + 9×4] = 5 × (6 + 36) = 5 × 42 = 210

Geometric Progression (G.P.)

• A sequence where each term is obtained by multiplying the previous term by a constant ratio, r.
• General term:
  uₙ = a × rⁿ⁻¹
• Sum of first n terms (r ≠ 1):
  Sₙ = a × (1 − rⁿ) / (1 − r)
• Sum to infinity (if |r| < 1):
  S∞ = a / (1 − r)

Example:

Find the 5th term and sum of the first 5 terms of: 8, 4, 2, …

• a = 8, r = 0.5
• u₅ = 8 × (0.5)⁴ = 8 × 1/16 = 0.5
• S₅ = 8 × (1 − 0.5⁵) / (1 − 0.5) = 8 × (1 − 1/32) / 0.5 = 8 × (31/32) / 0.5 = (248/32) × 2 = 15.5

12.4 Convergence of a Geometric Progression

• A G.P. converges only if |r| < 1
• If it does, you can use S∞ = a / (1 − r)

Example:

Does the G.P. 100, 50, 25, … converge?

• r = 0.5 < 1 ⇒ Converges
• S∞ = 100 / (1 − 0.5) = 100 / 0.5 = 200

12.5 Important Notes and Tips

• In the binomial expansion:
  • Number of terms is n + 1
  • Powers of a decrease, powers of b increase
  • Coefficients are symmetric
• For geometric series:
  • Diverges if |r| ≥ 1
  • Use proper signs in calculating a, r
• Use visual sketching to compare:
  • Arithmetic sequences → straight-line graphs
  • Geometric sequences → exponential curves

Symbols & Notation Refresher