Taylor Series Formula

Taylor series is a representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point: f(x) = Σ_(n=0)^∞ (f^(n)(a)/n!)(x-a)^n.

The Formula

f(x)=∑n=0∞f(n)(a)n!(x−a)n=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+⋯

When to use: Approximate any smooth function with a polynomial by matching the function's value, slope, curvature, and all higher derivatives at a single point. The more terms you include, the wider the region where the polynomial closely matches the function. It's like fitting a polynomial glove onto the function's hand.

Quick Example

ex=∑n=0∞xnn!=1+x+x22!+x33!+⋯
e1≈1+1+0.5+0.167+0.042+⋯=2.718…
Each term improves the approximation.

Notation

Tn(x) = Taylor polynomial of degree n. Rn(x) = remainder (error) term.

What This Formula Means

A representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point: f(x)=∑n=0∞f(n)(a)n!(x−a)n
When a=0, it's called a Maclaurin series.

Approximate any smooth function with a polynomial by matching the function's value, slope, curvature, and all higher derivatives at a single point. The more terms you include, the wider the region where the polynomial closely matches the function. It's like fitting a polynomial glove onto the function's hand.

Formal View

f(x)=∑n=0∞f(n)(a)n!(x−a)n for ∣x−a∣<R (radius of convergence). Taylor's theorem with remainder: f(x)=Tn(x)+Rn(x) where Rn(x)=f(n+1)(c)(n+1)!(x−a)n+1 for some c between a and x (Lagrange form).

Worked Examples

Example 1

easy
Find the Maclaurin series for ex up to the x4 term.

Answer

ex=1+x+x22+x36+x424+⋯

First step

1
The Taylor series formula is f(x)=∑n=0∞f(n)(0)n!xn. Compute the derivatives of ex at x=0.

Full solution

  1. 2
    Since every derivative of ex is ex, we have f(n)(0)=e0=1 for all n.
  2. 3
    Substitute into the formula: ex=∑n=0∞xnn!=1+x+x22!+x33!+⋯
Every derivative of ex is ex, so all coefficients are 1/n!. Converges for all x.

Example 2

hard
Find the Maclaurin series for ln⁡(1+x) and state the interval of convergence.

Example 3

medium
Approximate e0.2 using the Maclaurin series for ex through the x3 term.

Common Mistakes

  • Forgetting the n! in the denominator - each coefficient is f(n)(a)n!, not just f(n)(a).
  • Assuming convergence everywhere - the series only represents f within its interval of convergence; check it.
  • Centering at the wrong point - use (x−a) powers about the center a; Maclaurin specifically means a=0.

Why This Formula Matters

It is how calculators and computers evaluate transcendental functions, and it turns intractable integrals and limits into polynomial arithmetic. Conceptually it unifies all of differential calculus — the whole local behavior of a function is encoded in its derivatives at a single point. Recognizing it by "Am I building an infinite polynomial whose successive derivatives at one center match the function's?" — rather than by familiar numbers — is what lets a student tell it apart from power series (general) and linear approximation / tangent line and infinite geometric series in a mixed problem set.

Frequently Asked Questions

What is the Taylor Series formula?

A representation of a function as an infinite sum of terms calculated from the function's derivatives at a single point: f(x)=∑n=0∞f(n)(a)n!(x−a)n
When a=0, it's called a Maclaurin series.

How do you use the Taylor Series formula?

Approximate any smooth function with a polynomial by matching the function's value, slope, curvature, and all higher derivatives at a single point. The more terms you include, the wider the region where the polynomial closely matches the function. It's like fitting a polynomial glove onto the function's hand.

What do the symbols mean in the Taylor Series formula?

Tn(x) = Taylor polynomial of degree n. Rn(x) = remainder (error) term.

Why is the Taylor Series formula important in Math?

It is how calculators and computers evaluate transcendental functions, and it turns intractable integrals and limits into polynomial arithmetic. Conceptually it unifies all of differential calculus — the whole local behavior of a function is encoded in its derivatives at a single point. Recognizing it by "Am I building an infinite polynomial whose successive derivatives at one center match the function's?" — rather than by familiar numbers — is what lets a student tell it apart from power series (general) and linear approximation / tangent line and infinite geometric series in a mixed problem set.

What do students get wrong about Taylor Series?

The procedure for taylor series is the easy part; the trap is forgetting the n! in the denominator. Asking "Am I building an infinite polynomial whose successive derivatives at one center match the function's?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Taylor Series formula?

Before studying the Taylor Series formula, you should understand: derivative, differentiation rules, infinite geometric series, convergence divergence.

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