When you ask a computer or a calculator for the sine of an angle, or e to some power, it is using an algorithm to make this approximation. Many such algorithms are based on infinite series. An infinite series is a sum of infinitely many terms.

The infinite series converges for the following rules :

Geometric Series is a series of the form : Sn = a + ar + ar² +... + ar^n-1 = a(1-rⁿ)/(1-r) where a is the first term, r is the common ratio, and n is the number of terms. The sum can also be represented by the formula in the figure below:

If |r| < 1, lim x->inf of rⁿ = 0, thus:

By integrating the geometric series, we got -ln(1 - x) = x + x²/2 + x³/3 + x⁴/4 + ...

By substituting -x for x, we got ln(1 + x) = x - x²/2 + x³/3 - x⁴/4 +...

The major question still dangling is this: Given a function, can we represent it as a power series in x, or more generally, in x- a? More precisely, can we find numbers C₀,C₁,C₂,C₃,... such that:

f(x) = c₀ + c₁(x - a) + c₂(x - a)² + c₃(x - a)³ +....

If such representation exists, by substituting x = a, we get :

c₀ = f(a)

c₁ = f'(a)

c₂ = f''(a)/(2!)

thus: c_n = f^[n](a)/(n!)

To make it easier to evaluate, use MacLaurin series, which is a Taylor series at a = 0:

Select the functions: y = 1/1-x y = ln(1-x) y = e^x y = sin(x) y = cos(x)

 

x: 0

Number of terms: 4

y approx = -, y exact = -, err = -

Legend : Red is the Maclaurin approximation, Blue is the real function.

Using that properties we can now approximate many trancedental functions. With error function :