The Pension Fund Puzzle

Formula: (1 + 1/n)^n. Limit as n → ∞: Euler's number e ≈ 2.71828. One rupee at 100% continuous compound interest grows to exactly e rupees after one year — no matter how frequently you compound, you cannot exceed this ceiling.

This is the definition of Euler's number e through the compound interest limit. The intuition: more frequent compounding means more interest-on-interest, but there is a ceiling — you cannot extract infinite growth from a fixed rate by compounding more often. Step 1: Compound interest formula: value = P×(1 + r/n)^(n×t). For P=1, r=1 (100%), t=1 year: value = (1 + 1/n)^n. Step 2: Test finite values of n: n=1 (annual): (1+1)^1 = 2. n=2 (semi-annual): (1.5)^2 = 2.25. n=4 (quarterly): (1.25)^4 ≈ 2.441. n=12 (monthly): (1+1/12)^12 ≈ 2.613. n=365 (daily): ≈ 2.7146. n=8760 (hourly): ≈ 2.7181. Step 3: The sequence converges — slowly but surely — toward a fixed value. It never reaches 3. The limit is defined as e = lim(n→∞) of (1+1/n)^n. Step 4: e ≈ 2.71828182845904523536... It is irrational and transcendental — it cannot be expressed as a fraction or as the root of any polynomial with rational coefficients. Step 5: e appears in: continuous growth and decay, the natural logarithm (ln), probability theory, complex analysis, and countless areas of physics and engineering. Key Insight: e is not just a mathematical curiosity — it is the natural base of exponential growth. Any process that grows proportionally to its current size (populations, radioactive decay, bank interest) is described by powers of e. The compound interest limit is the most intuitive way to understand why e is 'natural'.