Continuous Compounding

Pre-Calculus · Axiom Academy

As you compound more and more often, the balance climbs toward a ceiling — and that ceiling is A = Pe^ rt . 1. From Discrete to Continuous Recall the ordinary compound-interest formula, where n is the number of times interest is applied per year: Compound n times per year (discrete) What happens as we compound more and more often — quarterly, monthly, daily, every second ( )? The balance does not explode; it creeps toward a ceiling : The limit IS continuous compounding Why does e appear at all? Push the compound formula to its extreme with P=1 , r=1 , t=1 . You are left with one of the most famous limits in mathematics: This limit DEFINES the natural base The sequence climbs as n increases but settles onto a single value — never quite reaching it. Watch the staircase level off onto the line y = e : Putting the rate r and time t back in turns this limit into the clean formula for continuous growth: 3. Comparing Compounding Methods 1,000 invested at 8% for 5 years, under four compounding frequencies. The bars grow to their final balances — continuous reaches the ceiling, the others land just below it: 4. Visualizing Continuous Growth Plot the balance over time. Discrete compounding jumps in steps — a flat stretch, then a sudden bump when interest posts. Continuous compounding is the smooth curve those steps hug ever more tightly: 2,500 is invested at 5.5% annual interest, compounded continuously . Find the balance after 8 years.

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