Power Series Calculator – Taylor and Maclaurin Series
Calculate power series expansions for common functions
Table of Contents
How to Use
- Select the function type (exponential, sine, cosine, ln, or geometric)
- Enter the center point (0 for Maclaurin series)
- Specify the number of terms to calculate
- Enter the value at which to evaluate the series
- Click calculate to see the series expansion and approximation
What is a Power Series?
A power series is an infinite series of the form Σ aₙ(x-a)ⁿ, where aₙ are the coefficients, x is the variable, and a is the center point. Power series are used to represent functions as infinite sums of polynomial terms.
When the center point a = 0, the series is called a Maclaurin series. When a ≠ 0, it's called a Taylor series centered at a.
Common Power Series
| Function | Power Series | Convergence |
|---|---|---|
| e^x | Σ xⁿ/n! | All real x |
| sin(x) | Σ (-1)ⁿx^(2n+1)/(2n+1)! | All real x |
| cos(x) | Σ (-1)ⁿx^(2n)/(2n)! | All real x |
| ln(1+x) | Σ (-1)^(n+1)xⁿ/n | -1 < x ≤ 1 |
| 1/(1-x) | Σ xⁿ | |x| < 1 |
Taylor Series Formula
The Taylor series of a function f(x) centered at a is:
f(x) = Σ f⁽ⁿ⁾(a)/n! × (x-a)ⁿ for n = 0 to ∞
Where f⁽ⁿ⁾(a) is the nth derivative of f evaluated at x = a.
Convergence and Radius
Every power series has a radius of convergence R, which determines where the series converges:
- The series converges absolutely for |x - a| < R
- The series diverges for |x - a| > R
- At |x - a| = R, convergence must be tested separately
- R can be found using the ratio test or root test
Applications of Power Series
- Approximating function values
- Solving differential equations
- Evaluating limits and integrals
- Numerical analysis and computation
- Physics and engineering calculations
- Signal processing and Fourier analysis
- Computer graphics and animation
Frequently Asked Questions
- What's the difference between Taylor and Maclaurin series?
- A Maclaurin series is a special case of a Taylor series where the center point is 0. Taylor series can be centered at any point a, while Maclaurin series are always centered at x = 0.
- How many terms do I need for a good approximation?
- It depends on the function and how close x is to the center point. Generally, more terms give better accuracy. For most practical purposes, 5-10 terms provide good approximations near the center.
- Why does the series sometimes give wrong values?
- Power series only converge within their radius of convergence. For example, ln(1+x) only converges for -1 < x ≤ 1, so evaluating at x = 2 will give incorrect results.
- Can I use power series for any function?
- Not all functions have power series representations. A function must be infinitely differentiable at the center point to have a Taylor series. Some functions, like |x|, don't have power series at certain points.