How to Calculate Power Series
What is Power Series?
A power series represents a function as an infinite sum of terms involving powers of x. They are used to approximate functions like eˣ, sin(x), and cos(x) with polynomials.
Formula
General form: Σₙ₌₀^∞ aₙ(x−c)ⁿ = a₀ + a₁(x−c) + a₂(x−c)² + ...
- aₙ
- coefficient of the nth term
- x
- variable
- c
- center of the series
- n
- term index
Step-by-Step Guide
- 1eˣ = 1 + x + x²/2! + x³/3! + ...
- 2sin(x) = x − x³/3! + x⁵/5! − ...
- 3cos(x) = 1 − x²/2! + x⁴/4! − ...
- 4More terms = better approximation
Worked Examples
Input
eˣ at x=1, 5 terms
Result
1+1+0.5+0.167+0.042 ≈ 2.708 (actual: 2.718)
Input
sin(π/6), 4 terms
Result
≈ 0.5 (exact)
Frequently Asked Questions
What is the radius of convergence?
The radius R is the distance from center c where the series converges. Use ratio or root test to find R.
What is a Taylor series?
A Taylor series is a power series centered at a point c using derivatives: f(x) = Σ f⁽ⁿ⁾(c)/n! (x−c)ⁿ.
What is a Maclaurin series?
A Taylor series centered at c = 0. Example: eˣ = Σ xⁿ/n!.
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