Taylor Series

A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point.

Generic Formula

f(x) = f(a) + f'(a)(x - a)/1! + f''(a)(x - a)²/2! + ...

Common Taylor Series Examples

• e^x = 1 + x/1! + x²/2! + x³/3! + ...
• a^x = 1 + x·ln(a)/1! + (x·ln(a))²/2! + (x·ln(a))³/3! + ...
• ln(1 + x) = x - x²/2 + x³/3 - x⁴/4 + x⁵/5 + ...
• sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ...
• cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...
• tan(x) = x + x³/3 + 2x⁵/15 + 17x⁷/315 + ...