Survey

* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project

Document related concepts
no text concepts found
Transcript
```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)ππ‘ .
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)ππ‘ = π[(1 + π) ]ππ‘ .
Now, as π β β, π β β
1 π
π
1 π
π
β΄ π[(1 + ) ]ππ‘ becomes π[ lim (1 + ) ]ππ‘ , which is ππ ππ‘ .
πββ
This is the formula for continuous compound interest where e is Eulerβs number.
1 π
Aim: To prove lim (1 + π) = π.
πββ
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.
πββ
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
```
Related documents