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```Continuous compound interest and the Maclaurin Series expansion
Background
The compound interest formula is 𝐴 = 𝑃(1 + 𝑟)𝑡 , where P is the principal, r is the annual interest
rate and t is time in years.
𝑟
For n compounds per year we have 𝐴 = 𝑃(1 + 𝑛)𝑛𝑡 .
Now, what happens as 𝑛 → ∞?, we end up with continuous compound interest.
𝑛
1 𝑚
𝑟
Q1. By letting 𝑟 = 𝑚, show that 𝑃(1 + 𝑛)𝑛𝑡 = 𝑃[(1 + 𝑚) ]𝑟𝑡 .
Now, as 𝑛 → ∞, 𝑚 → ∞
1 𝑚
𝑚
1 𝑚
𝑚
∴ 𝑃[(1 + ) ]𝑟𝑡 becomes 𝑃[ lim (1 + ) ]𝑟𝑡 , which is 𝑃𝑒 𝑟𝑡 .
𝑚→∞
This is the formula for continuous compound interest where e is Euler’s number.
1 𝑚
Aim: To prove lim (1 + 𝑚) = 𝑒.
𝑚→∞
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 + 𝑥)𝑛 .
1
𝑛
ii) Hence find the Maclaurin series expansion for (1 + )𝑛 .
1 𝑛
Q4. Now find lim (1 + 𝑛) . Show that this is the same as your result in Q2.
𝑛→∞
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 –
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
```