Taylor Series Calculator

f(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)² + (f'''(a)/3!)(x-a)³ + ...

Or more formally:

f(x) = Σ(n=0 to ∞) [f⁽ⁿ⁾(a)/n!](x-a)ⁿ

where:

• f(x) is the function being expanded
• a is the center point of the expansion
• f⁽ⁿ⁾(a) is the nth derivative at point a
• n! is the factorial of n

A Maclaurin series is a special case of Taylor series centered at a = 0:

f(x) = f(0) + f'(0)x + (f''(0)/2!)x² + (f'''(0)/3!)x³ + ...

Common Maclaurin Series:

• e^x = 1 + x + x²/2! + x³/3! + ...
• sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ...
• cos(x) = 1 - x²/2! + x⁴/4! - x⁶/6! + ...
• ln(1+x) = x - x²/2 + x³/3 - x⁴/4 + ... (|x| < 1)

Radius of Convergence

The series converges for |x-a| < R, where R is the radius of convergence

Error Bound (Lagrange Error)

|R_n(x)| ≤ (M|x-a|ⁿ⁺¹)/((n+1)!(1-|x-a|/R))

where M is the maximum value of |f⁽ⁿ⁺¹⁾(t)| on [a,x]

Common Convergence Intervals

• e^x: Infinite radius
• sin(x), cos(x): Infinite radius
• ln(1+x): |x| < 1
• 1/(1-x): |x| < 1