Binomial Theorem

Divisibility, Approximations and Applications

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Divisibility, Approximations and Applications

Divisibility, Approximations and Applications

This lesson covers the following key concepts:

  • Divisibility problems using binomial theorem
  • Finding remainders
  • Approximations using (1 + x)ⁿ for small x
  • Exponential series eˣ
  • Logarithmic series
  • Binomial theorem for negative/fractional indices
  • Condition: |x| < 1 for infinite expansion

Important Formulas

  • Remainder: write a=b±k, expand (b±k)n\text{Remainder: write } a = b \pm k, \text{ expand } (b \pm k)^n
  • (1+x)n≈1+nx for small x(1 + x)^n \approx 1 + nx \text{ for small } x
  • ex=1+x+x22!+x33!+...e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ...
  • ln⁡(1+x)=x−x22+x33−... for ∣x∣<1\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - ... \text{ for } |x| < 1
  • (1+x)n=1+nx+n(n−1)2!x2+... for ∣x∣<1,n∈Q(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + ... \text{ for } |x| < 1, n \in \mathbb{Q}
  • (1+x)−1=1−x+x2−x3+...(1 + x)^{-1} = 1 - x + x^2 - x^3 + ...