Exponential & Logarithmic Series & Mathematical Induction 

Exponential & Logarithmic Series

1) The number e

• Definition: e = 1 + 1/1! + 1/2! + 1/3! + …
• Also: e = lim_n→∞(1 + 1/n)ⁿ
• Series form: e = Σ_n=0^∞ (1/n!)
• Notes:
  • 2 < e < 3, approximate value e ≈ 2.718281828
  • e is irrational (e ∉ Q)

2) Exponential Series

• For real x: e^x = 1 + x + x²/2! + x³/3! + … = Σ_n=0^∞ xⁿ/n!
• Exponential theorem (a > 0): a^x = 1 + x(log a) + (x log a)²/2! + (x log a)³/3! + …
  i.e., a^x = Σ_n=0^∞ (x log a)ⁿ/n!

Standard Deductions

• Replace x by −x: e^−x = 1 − x + x²/2! − x³/3! + … = Σ_n=0^∞ (−1)ⁿ xⁿ/n!
• Put x = 1: e = 1 + 1/1! + 1/2! + 1/3! + … = Σ_n=0^∞ 1/n!
• Put x = −1: e⁻¹ = 1 − 1/1! + 1/2! − 1/3! + … = Σ_n=0^∞ (−1)ⁿ/n!
• (e^x + e^−x)/2 = 1 + x²/2! + x⁴/4! + x⁶/6! + …
  i.e., (e^x + e^−x)/2 = Σ_n=0^∞ x²ⁿ/(2n)!
• Put x = 1: (e + e⁻¹)/2 = 1 + 1/2! + 1/4! + 1/6! + … = Σ_n=0^∞ 1/(2n)!

3) Logarithmic Series

• For |x| < 1: log(1 + x) = x − x²/2 + x³/3 − x⁴/4 + …
  i.e., log(1 + x) = Σ_n=1^∞ (−1)ⁿ⁻¹ xⁿ/n

Standard Deductions (Log Series)

• log(1 − x) = −x − x²/2 − x³/3 − x⁴/4 − … = −Σ_n=1^∞ xⁿ/n
• log(1 + x) − log(1 − x) = log((1 + x)/(1 − x)) = 2(x + x³/3 + x⁵/5 + …)
• log(1 + x) + log(1 − x) = log(1 − x²) = −(x² + x⁴/2 + x⁶/3 + …)
• Natural log to common log: log₁₀(N) = log_e(N) × 0.43429448
• log 2 = 1 − 1/2 + 1/3 − 1/4 + … (also written as log 2 = 1/(2·1) + 1/(3·4) + 1/(5·6) + …)

Mathematical Induction

Mathematical Statement

• Statements involving mathematical relations are called mathematical statements.
• Examples: “2 divides 16”, “(x + 1) is a factor of x² − 3x + 2”.

Principle of Mathematical Induction (PMI)

• First Principle (P(n) over natural numbers):
  • Step 1: Prove P(1) is true.
  • Step 2: Assume P(m) is true and prove P(m + 1) is true.
  • Conclusion: Then P(n) is true for all natural numbers n.
• Second Principle:
  • Step 1: Prove P(1) is true.
  • Step 2: Prove P(m + 1) is true assuming P(n) is true for all n with 1 ≤ n ≤ m.
  • Conclusion: Then P(n) is true for all natural numbers.