What Is a Taylor Series?
A Taylor series is an infinite polynomial expansion that represents a smooth function $f(x)$ as a sum of terms built from the function's derivatives at a single center point $a$. It is one of the most powerful tools in calculus — letting us approximate complex functions (like $\sin x$ or $e^x$) with simple polynomials that are easy to compute, differentiate, and integrate.
The general Taylor series formula centered at $a$:
Taylor vs. Maclaurin Series
A Maclaurin series is simply a Taylor series centered at $a = 0$. Every Maclaurin series is a Taylor series, but not vice versa. Set center = 0 in this calculator to get the Maclaurin expansion; use any other value of $a$ for a general Taylor expansion centered at that point.
Taylor Series (any center $a$)
Use when approximating near a non-zero point, or when $f$ is singular at 0 (e.g., $\ln x$ expanded about $a=1$).
Maclaurin Series ($a = 0$)
Most common for $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$, $\arctan x$. Simplest notation, most widely tabulated.
Taylor Polynomial $T_n(x)$
The finite truncation using the first $n+1$ terms. Used in numerical methods, error estimation, and engineering approximations.
Remainder Term $R_n(x)$
The error: $|R_n(x)| \leq \frac{M|x-a|^{n+1}}{(n+1)!}$ where $M$ bounds $|f^{(n+1)}|$. More terms = smaller error.
Essential Taylor Series — Reference Table
These are the most important Maclaurin series every calculus student must know. Our calculator derives all of these with full steps and interactive graphs:
| Function f(x) | Maclaurin Series Expansion | Radius R |
|---|---|---|
| e^x | $1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots = \displaystyle\sum_{n=0}^{\infty}\dfrac{x^n}{n!}$ | $\infty$ |
| sin(x) | $x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots = \displaystyle\sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n+1}}{(2n+1)!}$ | $\infty$ |
| cos(x) | $1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots = \displaystyle\sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n}}{(2n)!}$ | $\infty$ |
| ln(1+x) | $x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \cdots = \displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}x^n}{n}$ | $1$ |
| arctan(x) | $x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \cdots = \displaystyle\sum_{n=0}^{\infty}\dfrac{(-1)^n x^{2n+1}}{2n+1}$ | $1$ |
| 1/(1-x) | $1 + x + x^2 + x^3 + \cdots = \displaystyle\sum_{n=0}^{\infty}x^n$ | $1$ |
| sqrt(1+x) | $1 + \dfrac{x}{2} - \dfrac{x^2}{8} + \dfrac{x^3}{16} - \cdots$ (Binomial) | $1$ |
| sinh(x) | $x + \dfrac{x^3}{3!} + \dfrac{x^5}{5!} + \cdots = \displaystyle\sum_{n=0}^{\infty}\dfrac{x^{2n+1}}{(2n+1)!}$ | $\infty$ |
How to Find a Taylor Series — Step by Step
- Choose the center point $a$ (usually $a=0$ for Maclaurin series, otherwise the point of approximation)
- Compute successive derivatives: $f(a),\; f'(a),\; f''(a),\; f'''(a), \ldots, f^{(n)}(a)$
- Divide each by the factorial: coefficient of the $n$-th term is $\dfrac{f^{(n)}(a)}{n!}$
- Build the polynomial: $T_n(x) = \displaystyle\sum_{k=0}^{n}\dfrac{f^{(k)}(a)}{k!}(x-a)^k$
- Determine convergence: use the Ratio Test to find the radius of convergence $R$
Our calculator automates all five steps. The interactive graph shows exactly how well $T_n(x)$ approximates $f(x)$ — click "Add 2 More Terms" to watch accuracy improve in real time.
Applications of Taylor Series
🧮 Numerical Computing
All scientific calculators evaluate $\sin x$, $e^x$, $\ln x$ using Taylor polynomials internally. IEEE floating-point arithmetic is built on series approximations.
🔬 Physics & Engineering
The small-angle approximation $\sin\theta \approx \theta$ is the first-degree Maclaurin term. Fourier series, quantum mechanics, and general relativity all rely on Taylor expansions.
📉 Limits & L’Hôpital
Indeterminate forms like $0/0$ and $\infty/\infty$ are often resolved by expanding numerator and denominator as Taylor series and identifying the dominant term.
🤖 Machine Learning
Gradient descent uses the first-order Taylor approximation. Newton’s method uses the second-order (Hessian). Backpropagation relies on differentiability of each layer.
📊 Error Estimation
Taylor’s Remainder Theorem gives explicit bounds on approximation error. Essential for verifying numerical precision meets engineering or scientific tolerances.
🧲 Differential Equations
Power series solutions to ODEs use the Taylor framework. Bessel, Legendre, and Airy functions are expressed as Taylor-type series with no closed-form alternatives.
Frequently Asked Questions
A Taylor series is an infinite sum: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$. A Taylor polynomial of degree $n$ is the finite truncation using the first $n+1$ terms: $T_n(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!}(x-a)^k$. The polynomial is used for practical approximation; the full infinite series converges exactly to $f(x)$ within the radius of convergence.
Use the Ratio Test: compute $L = \lim_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|$ where $a_n$ are the series coefficients. The radius of convergence is $R = 1/L$. If $L=0$, then $R=\infty$ (converges everywhere). Standard results: $e^x$, $\sin x$, $\cos x$ have $R=\infty$; $\ln(1+x)$, $\arctan x$, $\frac{1}{1-x}$ have $R=1$.
Because $\sin x$ is an odd function: $\sin(-x) = -\sin(x)$. All even-order derivatives of $\sin x$ evaluated at 0 are zero (since $\sin(0) = 0$, $\sin''(0) = -\sin(0) = 0$, etc.), so those terms vanish. Similarly, $\cos x$ is even, so its Maclaurin series contains only even powers. General rule: odd functions have series with only odd powers; even functions have only even powers.
Yes — but not centered at 0, since $\ln(0)$ is undefined. Center it at $a=1$ instead: $\ln x = (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \cdots$, converging for $0 < x \leq 2$. Alternatively, use $\ln(1+x)$ centered at $a=0$, valid for $|x| < 1$. Both are fully supported in this calculator.
It depends on how far $x$ is from the center $a$ and how much accuracy you need. For $\sin x \approx x$ (degree 1), error is $\approx x^3/6$. For $x$ near 0, degree 3 gives very good accuracy. For larger $x$ or tighter tolerance, use degree 7–11. The interactive graph in this calculator shows the approximation visually — click "Add 2 More Terms" to increase accuracy interactively and watch the approximation improve.