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Continuous compound interest and the Maclaurin Series expansion Background The compound interest formula is π΄ = π(1 + π)π‘ , where P is the principal, r is the annual interest rate and t is time in years. π For n compounds per year we have π΄ = π(1 + π)ππ‘ . Now, what happens as π β β?, we end up with continuous compound interest. π 1 π π Q1. By letting π = π, show that π(1 + π)ππ‘ = π[(1 + π) ]ππ‘ . Now, as π β β, π β β 1 π π 1 π π β΄ π[(1 + ) ]ππ‘ becomes π[ lim (1 + ) ]ππ‘ , which is ππ ππ‘ . πββ This is the formula for continuous compound interest where e is Eulerβs number. 1 π Aim: To prove lim (1 + π) = π. πββ One way to do this is to use the Maclaurin series expansion. Q2. Find the Maclaurin series expansion for e. Q3. i) Find the Maclaurin series expansion for (1 + π₯)π . 1 π ii) Hence find the Maclaurin series expansion for (1 + )π . 1 π Q4. Now find lim (1 + π) . Show that this is the same as your result in Q2. πββ Q5. What will $1000 grow to in 1 year if it is compounded continuously at 3% p.a.? Compare this to $1000 for 1 year at 3% p.a. compounded monthly. Criteria assessed β 1. communicate mathematical ideas and information 2. analysis: demonstrate mathematical reasoning, analysis and strategy in problem solving situations 3. plan, organise and complete mathematical tasks