A Taylor series approximates a function by adding terms built from its mathematical pattern. For sine, it turns a calculation that once required tables into a repeatable program. section 3.1
The summation behaves like a loop: iterate across a range, calculate each term, and add it to a running total. The exact sine definition uses infinitely many terms, so a program chooses a finite n. section 3.4.1
Implementing sine in Julia
A compact Julia function sets n = 5, loops from 0 to n, and accumulates (-1)^i*x^(2i+1)/factorial(2i + 1). Its result closely matches Julia’s built-in sin: sine(0.5) returns 0.4794255386041834, while sin(0.5) returns 0.479425538604203. section 3.4.1
A practical Taylor-series workflow:
- Express the function as a summation. section 3.4.1
- Translate the summation into a for loop with a running total. section 3.4.1
- Increase n and compare the approximation with the built-in function. section 3.4.1
Listing 3.11 Sine implemented using the Taylor series
function sine(x)
n = 5
total = 0
for i in 0:n
total += (-1)^i*x^(2i+1)/factorial(2i + 1)
end
total
end
View the complete Julia implementation that converts the sine Taylor series into executable code. section 3.4.1
