Taylor Series Calculator
Instantly calculate Taylor and Maclaurin series expansions with step-by-step logic. Visualize how polynomials approximate complex functions.
Taylor Series Calculator
Supported: sin, cos, tan, exp, log, sqrt, etc.
0 for Maclaurin Series
Comprehensive Guide to Taylor Series
What is a Taylor Series?
In mathematics, a Taylor Series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. It is a powerful tool in calculus used to approximate complex, non-polynomial functions (such as eˣ, sin(x), ln(x)) using polynomials.
Polynomials are much easier to handle computationally—they can be easily integrated, differentiated, and evaluated by computers. This is why Taylor Series are fundamental to how calculators (and this very tool) compute values like sin(34°) or e²·⁵.
A Brief History
The concept was introduced by the English mathematician Brook Taylor in 1715. However, special cases of the series were known to Indian mathematicians in the 14th century (Madhava of Sangamagrama) and later by Gregory in the 17th century. The special case where the expansion is centered at zero (a=0) is named after the Scottish mathematician Colin Maclaurin, who made extensive use of this special case in the 18th century.
The Taylor Series Formula
A function f(x) that is infinitely differentiable at a real number a is equivalent to the power series. The general Taylor Series equation is given by:
f(x) = f(a) + f'(a)(x-a) + [f''(a)/2!](x-a)² + [f'''(a)/3!](x-a)³ + ...Or in compact Sigma notation:
∑ [f⁽ⁿ⁾(a) / n!] (x-a)ⁿ- f⁽ⁿ⁾(a): The n-th derivative of the function evaluated at point a.
- n!: The factorial of n (Example: 3! = 3 × 2 × 1 = 6).
- (x-a)ⁿ: The polynomial term centered at a.
How to Calculate Manually (e^x Taylor Series)
Let's derive the Taylor series for e^x (specifically the Maclaurin series centered at a=0). This is one of the most classic examples of a Taylor series of e^x.
Step 1: Find Derivatives
- f(x) = e^x
- f'(x) = e^x
- f''(x) = e^x
- f'''(x) = e^x
Step 2: Evaluate at Center (a=0)
- f(0) = e^0 = 1
- f'(0) = e^0 = 1
- f''(0) = e^0 = 1
- f'''(0) = e^0 = 1
Step 3: Plug into Formula
Using the formula:
e^x ≈ 1 + (1/1!)x + (1/2!)x² + (1/3!)x³
Result: eˣ = 1 + x + x²/2 + x³/6 + ...
Taylor Series vs. Maclaurin Series
Taylor Series
Expands a function around any real number a.
∑ [f⁽ⁿ⁾(a) / n!] (x-a)ⁿUse when you need to approximate a function near a specific point (e.g., approximate ln(x) near x=1).
Maclaurin Series
A special case where the center is always a=0.
∑ [f⁽ⁿ⁾(0) / n!] xⁿUsually simpler. Ideal for functions like sin(x) or eˣ near the origin.
Why do we need this? (Real World Applications)
Taylor series are not just abstract math; they run the world:
- Physics: In solving pendulum motion, the approximation sin(θ) ≈ θ (the first term of the Taylor series) simplifies differential equations significantly for small angles.
- Computer Science: Computers cannot technically calculate transcendental functions like sine or cosine directly. They use Taylor polynomials (or variations like CORDIC) to compute these values to high precision using simple addition and multiplication.
- Engineering: Control systems often linearize non-linear functions using the first two terms of a Taylor Series to apply standard linear control theory.
- Finance: Used in quantitative finance to approximate option prices and risk metrics (e.g., Delta-Gamma approximations).
Cheat Sheet: Common Taylor Series
| Function | Maclaurin Series | Interval of Convergence |
|---|---|---|
| eˣ | 1 + x + x²/2! + x³/3! + ... | (-∞, ∞) |
| sin(x) | x - x³/3! + x⁵/5! - ... | (-∞, ∞) |
| cos(x) | 1 - x²/2! + x⁴/4! - ... | (-∞, ∞) |
| 1/(1-x) | 1 + x + x² + x³ + ... | (-1, 1) |
| ln(1+x) | x - x²/2 + x³/3 - ... | (-1, 1] |
| arctan(x) | x - x³/3 + x⁵/5 - ... | [-1, 1] |
Advanced: Taylor Remainder Theorem
How accurate is the approximation? The error Rn(x) (Remainder) is given by the Lagrange Error Bound:
Where M is the maximum absolute value of the (n+1)-th derivative on the interval between a and x. This formula is crucial for determining how many terms are needed to achieve a specific decimal precision.
Frequently Asked Questions
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