∑ Taylor Series Calculator: Complete Guide
Master Taylor and Maclaurin series expansions with our advanced calculator featuring visual comparisons and detailed coefficient calculations.
Taylor Series Formula
$$f(x) = \\sum_{n=0}^{\\infty} \\frac{f^{(n)}(a)}{n!}(x-a)^n$$
Where f^(n)(a) is the nth derivative evaluated at x=a
A Taylor series represents a function as an infinite sum of terms calculated from derivatives at a single point. When a=0, it's called a Maclaurin series.
Common Taylor Series Expansions
| Function | Taylor Series (Maclaurin) | Convergence |
|---|---|---|
| sin(x) | x - x³/3! + x⁵/5! - x⁷/7! + ... | All real x |
| cos(x) | 1 - x²/2! + x⁴/4! - x⁶/6! + ... | All real x |
| e^x | 1 + x + x²/2! + x³/3! + x⁴/4! + ... | All real x |
| ln(1+x) | x - x²/2 + x³/3 - x⁴/4 + ... | -1 < x ≤ 1 |
| 1/(1-x) | 1 + x + x² + x³ + x⁴ + ... | |x| < 1 |
Frequently Asked Questions
Q: What is a Taylor series?
A: A Taylor series is an infinite sum of polynomial terms that approximates a function near a specific point. Each term uses higher-order derivatives: f(x) ≈ f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... The more terms you include, the better the approximation becomes near the center point.
Q: What's the difference between Taylor and Maclaurin series?
A: A Maclaurin series is simply a Taylor series centered at a=0. All Maclaurin series are Taylor series, but not all Taylor series are Maclaurin series. Example: Maclaurin for sin(x) = x - x³/3! + x⁵/5! - ..., while Taylor centered at a=π would use (x-π) instead of x.
Q: How many terms do I need for accuracy?
A: It depends on desired accuracy and distance from center point. Near the center, few terms suffice. Farther away requires more terms. For sin(x) near x=0, just 3 terms (up to x⁵) gives excellent accuracy. Test different orders to see convergence visually using our graph feature.
Q: Do all functions have Taylor series?
A: No! Functions must be infinitely differentiable at the center point AND the series must converge to the function. Some functions like |x| don't have Taylor series at x=0 (not differentiable there). Others like e^(-1/x²) have all derivatives zero at x=0, giving a trivial series that doesn't represent the function.