Taylor Series Formula

Taylor series are 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^.

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+⋯f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots

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!+⋯e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots
e1≈1+1+0.5+0.167+0.042+⋯=2.718…e^1 \approx 1 + 1 + 0.5 + 0.167 + 0.042 + \cdots = 2.718\ldots
Each term improves the approximation.

Notation

Tn(x)T_n(x) = Taylor polynomial of degree nn. Rn(x)R_n(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)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
When a=0a = 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)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n for ∣x−a∣<R|x - a| < R (radius of convergence). Taylor's theorem with remainder: f(x)=Tn(x)+Rn(x)f(x) = T_n(x) + R_n(x) where Rn(x)=f(n+1)(c)(n+1)!(x−a)n+1R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1} for some cc between aa and xx (Lagrange form).

Worked Examples

Example 1

easy
Find the Maclaurin series for exe^x up to the x4x^4 term.

Answer

ex=1+x+x22+x36+x424+⋯e^x = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots

First step

1
The Taylor series formula is f(x)=∑n=0∞f(n)(0)n!xnf(x) = \sum_{n=0}^{\infty}\frac{f^{(n)}(0)}{n!}x^n. Compute the derivatives of exe^x at x=0x=0.

Full solution

  1. 2
    Since every derivative of exe^x is exe^x, we have f(n)(0)=e0=1f^{(n)}(0) = e^0 = 1 for all nn.
  2. 3
    Substitute into the formula: ex=∑n=0∞xnn!=1+x+x22!+x33!+⋯e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!} = 1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots
Every derivative of exe^x is exe^x, so all coefficients are 1/n!1/n!. Converges for all xx.

Example 2

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

Example 3

medium
Approximate e0.2e^{0.2} using the Maclaurin series for exe^x through the x3x^3 term.

Common Mistakes

  • Forgetting the n!n! in the denominator - each coefficient is f(n)(a)n!\frac{f^{(n)}(a)}{n!}, not just f(n)(a)f^{(n)}(a).
  • Assuming convergence everywhere - the series only represents ff within its interval of convergence; check it.
  • Centering at the wrong point - use (x−a)(x-a) powers about the center aa; Maclaurin specifically means a=0a=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)nf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n
When a=0a = 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)T_n(x) = Taylor polynomial of degree nn. Rn(x)R_n(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!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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