Taylor Series
Taylor Series - Solve mathematical problems with step-by-step solutions.
Taylor Series Calculator
Expand a function into a series
Taylor Series
A Taylor series is a representation of a function as an infinite sum of terms, calculated from the values of the function's derivatives at a single point. It's a way to approximate a function with a polynomial. When the expansion point is 0, it's called a Maclaurin series.
How the Taylor Series Calculator Works
The Taylor Series Calculator expands functions into infinite polynomial series, providing powerful approximations of complex functions near a specific point. Taylor series are fundamental in mathematical analysis, numerical methods, physics, and engineering, allowing us to approximate transcendental functions using only polynomial operations.
What is a Taylor Series?
A Taylor series represents a function f(x) as an infinite sum of terms calculated from the function's derivatives at a single point a:
f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)2/2! + f'''(a)(x-a)3/3! + ...
Or more compactly: f(x) = Σn₌0^∞ [fn(a)/n!]·(x-a)n
Special Cases
- Maclaurin Series: Taylor series centered at a = 0. Many common series are Maclaurin series.
- Power Series: General form Σanxn. Taylor series are power series with specific coefficients.
- Polynomial Approximation: Truncating after n terms gives the nth-degree Taylor polynomial, approximating f near a.
Common Taylor/Maclaurin Series
- e^x = 1 + x + x2/2! + x3/3! + x4/4! + ...
- 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 + ... (for |x| < 1)
- 1/(1-x) = 1 + x + x2 + x3 + ... (for |x| < 1)
Convergence and Radius
A Taylor series may converge for all x, or only for x within a certain radius of convergence R around the center point a. Within the radius of convergence, the series equals the function exactly. Outside this radius, the series may diverge or converge to a different value.
Examples
Taylor Series Examples
Example 1: Exponential Function
Problem: Find the Maclaurin series for e^x
Solution:
- All derivatives of e^x equal e^x
- At x=0: f(0) = f'(0) = f''(0) = ... = e0 = 1
- Formula: eˣ = Σn₌0^∞ (1/n!)xn
- Expanded: e^x = 1 + x + x2/2 + x3/6 + x4/24 + ...
- Radius of convergence: ∞ (converges for all x)
Example 2: Sine Function
Problem: Find the Maclaurin series for sin(x)
Solution:
- Derivatives cycle: sin, cos, -sin, -cos, sin, ...
- At x=0: sin(0)=0, cos(0)=1, -sin(0)=0, -cos(0)=-1, ...
- Only odd powers appear with alternating signs
- sin(x) = x - x3/3! + x5/5! - x7/7! + ...
- sin(x) = Σn₌0^∞ [(-1)n/(2n+1)!]x^(2n+1)
Example 3: Taylor Polynomial Approximation
Problem: Approximate sin(0.1) using 3rd degree Taylor polynomial
Solution:
- P3(x) = x - x3/6 (first two non-zero terms)
- P3(0.1) = 0.1 - (0.1)3/6 = 0.1 - 0.000167 ≈ 0.0998334
- Actual: sin(0.1) ≈ 0.0998334166...
- Error: less than 0.0000001 (excellent approximation!)
Example 4: Cosine Function
Problem: Find the Maclaurin series for cos(x)
Solution:
- Derivatives at 0: cos(0)=1, -sin(0)=0, -cos(0)=-1, sin(0)=0, ...
- Only even powers with alternating signs
- cos(x) = 1 - x2/2! + x4/4! - x6/6! + ...
- cos(x) = Σn₌0^∞ [(-1)n/(2n)!]x^(2n)
Example 5: Geometric Series
Problem: Find the Taylor series for 1/(1-x) at a=0
Solution:
- This is the geometric series formula
- All derivatives follow pattern: fn(0) = n!
- 1/(1-x) = 1 + x + x2 + x3 + x4 + ...
- Converges only for |x| < 1 (radius of convergence = 1)
- At x=0.5: 1/(1-0.5) = 2 = 1 + 0.5 + 0.25 + 0.125 + ...
Example 6: Natural Logarithm
Problem: Find the Maclaurin series for ln(1+x)
Solution:
- f(x) = ln(1+x), f(0) = 0
- f'(x) = 1/(1+x), f'(0) = 1
- f''(x) = -1/(1+x)2, f''(0) = -1
- Pattern emerges: ln(1+x) = x - x2/2 + x3/3 - x4/4 + ...
- Converges for -1 < x ≤ 1
Example 7: Taylor Series at Non-Zero Center
Problem: Find Taylor series for f(x) = ln(x) centered at a=1
Solution:
- f(1) = ln(1) = 0
- f'(x) = 1/x, f'(1) = 1
- f''(x) = -1/x2, f''(1) = -1
- f'''(x) = 2/x3, f'''(1) = 2
- ln(x) = (x-1) - (x-1)2/2 + (x-1)3/3 - (x-1)4/4 + ...
Tips & Best Practices
Tips for Working with Taylor Series
- Memorize Common Series: Know the series for e^x, sin(x), cos(x), 1/(1-x), and ln(1+x) by heart.
- Use Known Series: Instead of computing derivatives, substitute into known series. For e^(-x), use e^x series with -x.
- Check Radius of Convergence: Always determine where the series converges before using it for approximations.
- Even/Odd Functions: Even functions have only even powers, odd functions only odd powers in Maclaurin series.
- Truncation for Approximation: For practical calculations, use enough terms to achieve desired accuracy.
- Alternating Series Error: For alternating series, the error is less than the first omitted term.
- Operations on Series: You can add, subtract, multiply, and differentiate/integrate series term by term (within radius of convergence).
- Substitution Technique: To find series for f(g(x)), substitute the series for g(x) into the series for f(x).
- Compare with Polynomial: The nth Taylor polynomial is the best polynomial approximation of degree ≤ n at the center point.
- Practical Applications: Calculators and computers use Taylor series to compute trigonometric and exponential functions!
Frequently Asked Questions
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