Taylor Series Formulas: Complete Reference Guide

📅 Updated: December 29, 2025 ⏱️ 15 min read 📚 Formula Reference 🎯 Exam Essential

This comprehensive Taylor Series formula cheat sheet contains every essential expansion you'll need for calculus, engineering, and physics. Bookmark this page or download the PDF version for quick reference during exams and problem-solving sessions. All formulas include convergence intervals, common notations, and practical examples.

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Includes all formulas + convergence tables + example problems

📑 Formula Index

1. General Taylor Series Formula

The fundamental Taylor Series expansion of a function \( f(x) \) about point \( a \):

General Taylor Series
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \]

Where:

  • \( f^{(n)}(a) \) = nth derivative of f evaluated at x = a
  • \( n! \) = factorial of n (0! = 1, 1! = 1, 2! = 2, etc.)
  • \( (x-a)^n \) = power centered at a
  • The series converges within radius R from a
Maclaurin Series (a = 0)
\[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n \]

Special case: Taylor series centered at 0. Most common expansions are Maclaurin series.

💡 Memory Aid

The Taylor series formula follows a simple pattern: Take the nth derivative at a, divide by n!, multiply by (x-a)ⁿ, sum from n=0 to infinity. For Maclaurin series, simply set a=0.

2. Elementary Functions

These are the most fundamental Taylor series that every calculus student must know.

\( e^x \)
\[ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]

Convergence: All real x (\( R = \infty \))

Notes: All derivatives are \( e^x \), and \( e^0 = 1 \)

\( \frac{1}{1-x} \)
\[ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \cdots \]

Convergence: \( |x| < 1 \) (\( R = 1 \))

Notes: Geometric series formula, very useful for integration

\( (1+x)^k \)
\[ (1+x)^k = \sum_{n=0}^{\infty} \binom{k}{n} x^n \]

Convergence: \( |x| < 1 \) (general binomial)

Where: \( \binom{k}{n} = \frac{k(k-1)\cdots(k-n+1)}{n!} \)

3. Trigonometric Functions

Sine and cosine have particularly elegant Taylor series with alternating signs and factorial denominators.

\( \sin x \)
\[ \sin x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} \] \[ = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \]

Convergence: All real x (\( R = \infty \))

Pattern: Odd powers only, alternating signs

\( \cos x \)
\[ \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \] \[ = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \]

Convergence: All real x (\( R = \infty \))

Pattern: Even powers only, alternating signs

\( \tan x \)
\[ \tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \cdots \]

Convergence: \( |x| < \frac{\pi}{2} \) (\( R = \frac{\pi}{2} \))

Notes: Coefficients involve Bernoulli numbers

\( \arctan x \)
\[ \arctan x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \] \[ = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \]

Convergence: \( |x| \leq 1 \) (\( R = 1 \))

Special: At x=1, gives Leibniz formula for π/4

4. Hyperbolic Functions

Hyperbolic functions have Taylor series similar to trigonometric functions but without alternating signs.

\( \sinh x \)
\[ \sinh x = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{(2n+1)!} \] \[ = x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} + \cdots \]

Convergence: All real x (\( R = \infty \))

Relation: \( \sinh x = \frac{e^x - e^{-x}}{2} \)

\( \cosh x \)
\[ \cosh x = \sum_{n=0}^{\infty} \frac{x^{2n}}{(2n)!} \] \[ = 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \cdots \]

Convergence: All real x (\( R = \infty \))

Relation: \( \cosh x = \frac{e^x + e^{-x}}{2} \)

5. Logarithmic Functions

Logarithmic Taylor series have simple patterns but limited convergence intervals.

\( \ln(1+x) \)
\[ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^n}{n} \] \[ = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \]

Convergence: \( -1 < x \leq 1 \) (\( R = 1 \))

Special: At x=1: \( \ln 2 = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots \)

\( \ln\left(\frac{1+x}{1-x}\right) \)
\[ \ln\left(\frac{1+x}{1-x}\right) = 2\sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1} \] \[ = 2\left(x + \frac{x^3}{3} + \frac{x^5}{5} + \frac{x^7}{7} + \cdots\right) \]

Convergence: \( |x| < 1 \) (\( R = 1 \))

Use: Faster convergence than standard ln(1+x)

6. Binomial Series

The generalized binomial theorem provides Taylor series for fractional and negative powers.

General Binomial Series
\[ (1+x)^k = \sum_{n=0}^{\infty} \binom{k}{n} x^n \] \[ = 1 + kx + \frac{k(k-1)}{2!}x^2 + \frac{k(k-1)(k-2)}{3!}x^3 + \cdots \]

Convergence: \( |x| < 1 \) for k not positive integer

Where: \( \binom{k}{n} = \frac{k(k-1)\cdots(k-n+1)}{n!} \)

\( \sqrt{1+x} \) (k=½)
\[ \sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \frac{5x^4}{128} + \cdots \]

Convergence: \( |x| < 1 \)

\( \frac{1}{\sqrt{1+x}} \) (k=-½)
\[ \frac{1}{\sqrt{1+x}} = 1 - \frac{x}{2} + \frac{3x^2}{8} - \frac{5x^3}{16} + \frac{35x^4}{128} - \cdots \]

Convergence: \( |x| < 1 \)

7. Special Taylor Series

These series appear frequently in advanced mathematics and physics applications.

Error Function
\[ \operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{n!(2n+1)} \]

Convergence: All real x (\( R = \infty \))

Use: Probability, heat conduction

Bessel Function J₀
\[ J_0(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{2^{2n} (n!)^2} \]

Convergence: All real x (\( R = \infty \))

Use: Wave equations, cylindrical symmetry

Exponential Integral
\[ \operatorname{Ei}(x) = \gamma + \ln|x| + \sum_{n=1}^{\infty} \frac{x^n}{n \cdot n!} \]

Convergence: All x ≠ 0

Where: γ ≈ 0.5772 (Euler-Mascheroni constant)

8. Remainder Formulas

Error estimation for finite Taylor polynomials using different forms of the remainder term.

Taylor's Theorem with Remainder
\[ f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k + R_n(x) \]
Lagrange Form
\[ R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x-a)^{n+1} \]

Where: c between a and x

Use: General error bound

Integral Form
\[ R_n(x) = \frac{1}{n!} \int_a^x (x-t)^n f^{(n+1)}(t) \, dt \]

Use: Precise error calculation

Note: Requires integrability

Cauchy Form
\[ R_n(x) = \frac{f^{(n+1)}(c)}{n!} (x-c)^n (x-a) \]

Where: c between a and x

Use: Alternative to Lagrange

9. Convergence Reference Table

Quick reference for convergence radii of common Taylor series (centered at 0).

Function Series Radius (R) Converges For
\( e^x \) \( \sum \frac{x^n}{n!} \) ∞ All real x
\( \sin x \) \( \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!} \) ∞ All real x
\( \cos x \) \( \sum \frac{(-1)^n x^{2n}}{(2n)!} \) ∞ All real x
\( \ln(1+x) \) \( \sum \frac{(-1)^{n+1} x^n}{n} \) 1 \( -1 < x \leq 1 \)
\( \frac{1}{1-x} \) \( \sum x^n \) 1 \( |x| < 1 \)
\( \arctan x \) \( \sum \frac{(-1)^n x^{2n+1}}{2n+1} \) 1 \( |x| \leq 1 \)
\( (1+x)^k \) (k∉ℕ) \( \sum \binom{k}{n} x^n \) 1 \( |x| < 1 \)
\( \sinh x \) \( \sum \frac{x^{2n+1}}{(2n+1)!} \) ∞ All real x
\( \cosh x \) \( \sum \frac{x^{2n}}{(2n)!} \) ∞ All real x

10. Formula Usage Tips & Tricks

🎯 Pro Tips for Using Taylor Series Formulas
  • Combine series: Multiply or add known series to find new ones.
  • Substitution: Replace x with another expression (like -x² in eˣ to get e⁻ˣ²).
  • Differentiate/Integrate: Get new series by differentiating or integrating known ones.
  • Partial fractions: Break complex rational functions into simpler series.
  • Check convergence: Always verify radius of convergence for your specific x value.
  • Error estimation: Use remainder formulas to determine required terms for desired accuracy.
Example: Combining Series
Find Taylor series for f(x) = eˣ sin x using known series:

eˣ = 1 + x + x²/2! + x³/3! + x⁴/4! + ...
sin x = x - x³/3! + x⁵/5! - ...

Multiply term by term:
f(x) = (1)(x) + (1)(-x³/3!) + (x)(x) + (x)(-x³/3!) + (x²/2!)(x) + ...
= x - x³/6 + x² + (-x⁴/6) + x³/2 + ...
= x + x² + (1/2 - 1/6)x³ - x⁴/6 + ...
= x + x² + x³/3 - x⁴/6 + ...

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References & Further Reading

  • Abramowitz, M., & Stegun, I. A. (1964). Handbook of Mathematical Functions. Dover.
  • Gradshteyn, I. S., & Ryzhik, I. M. (2014). Table of Integrals, Series, and Products (8th ed.). Academic Press.
  • Weisstein, E. W. (2025). "Taylor Series." From MathWorld—A Wolfram Web Resource.
  • What is Taylor Series? - Conceptual understanding guide.
  • Wikipedia: List of Mathematical Series - Comprehensive series collection.