Key formulae

Geometric series Denote the initial value of the series as

For a series starting at i = 0 the i-th term is and the sum up to t is

For a series starting at i = 1 the i-th term is and the sum up to t is

The sum of an infinite geometric series, so long as is

Discretely compounded interest

Define as the principal and as the value of the investment at time t.

where r is the interest rate per period and t is the number of compoundings.

If the annual rate is r and there are m periods in the year then

If the rate per period is and the interest is compounded for t years then

Continuously compounded interest

A principal amount continuously compounded at an interest rate of for a time period t is worth at time t

Differentiation

The product rule:

The quotient rule:

The chain rule:

Integration

The integral of f(x) (area under the curve) with respect to x between the points x=a and x=b is

where F is the anti-derivative of f.