Equations · Calculus and analysis
Taylor series
A function rebuilt from its derivatives at one point, as an infinite sum of powers.
f(x) = ∑ₙ₌₀^∞ f⁽ⁿ⁾(a) (x − a)ⁿ / n!
Aethel says
f of x equals the sum from n equals 0 to infinity of the nth derivative of f at a times the quantity x minus a, to the power of n, over n factorial
What it says
Near the point a, a smooth function behaves like a polynomial. The Taylor series uses the value of the function and all its derivatives at a to write that polynomial out, term by term.
Keeping only the first few terms gives approximations such as sin x being close to x for small x. With a equal to zero the series is called a Maclaurin series.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| f⁽ⁿ⁾(a) | the nth derivative of f at a | f differentiated n times, evaluated at a |
| (x − a)ⁿ | the quantity x minus a, to the power of n | the distance from a, raised to the power n |
| n! | n factorial | 1 × 2 × 3 × … × n, with 0! equal to 1 |
Where you meet it
Calculus and numerical analysis, and every approximation in physics and engineering.
Questions
How do you say Taylor series out loud?
f(x) = ∑ₙ₌₀^∞ f⁽ⁿ⁾(a) (x − a)ⁿ / n! is read: “f of x equals the sum from n equals 0 to infinity of the nth derivative of f at a times the quantity x minus a, to the power of n, over n factorial.” That is exactly what Aethel says when it meets the formula in a book.
What does f⁽ⁿ⁾(a) mean in Taylor series?
f⁽ⁿ⁾(a) (said “the nth derivative of f at a”) is f differentiated n times, evaluated at a.
What does (x − a)ⁿ mean in Taylor series?
(x − a)ⁿ (said “the quantity x minus a, to the power of n”) is the distance from a, raised to the power n.
Can I listen to the equations in my own textbook like this?
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