Taylor series calculator with steps

What this page covers

• Taylor polynomials of a chosen degree about any center $x = a$, and Maclaurin polynomials about 0.
• Each derivative evaluated at the center and divided by the matching factorial to give the coefficient.
• The standard series for $e^x$, $\sin x$, $\cos x$, $\ln(1+x)$ and $\frac{1}{1-x}$, and substitution into them.
• Not covered here: the radius and interval of convergence of the full series (use Power series calculator).

How to enter the problem

• Type the function, the center and the degree, such as Taylor polynomial of degree 3 of ln x at x = 1.
• Scan it. Fit the one problem in the frame.
• Choose a photo. Several problems in one photo are solved as a list.
• Then tap Solve.

Worked examples

Common mistakes

• Leaving out the factorials. For $e^{2x}$ that gives $1 + 2x + 4x^2 + 8x^3$ instead of $1 + 2x + 2x^2 + \frac{4x^3}{3}$.
• Writing powers of $x$ instead of $(x - a)$ when the center is not 0. About $x = 1$, every term uses $(x-1)^n$.
• Counting the degree by the number of terms. The degree 4 polynomial of $\cos x$ has only three nonzero terms, because the odd derivatives of $\cos x$ are 0 at the center.

Checks, assumptions and limits

• Check numerically near the center. At $x = 0.1$ the runnable example gives $1 - 0.005 + 0.0000041667 \approx 0.9950042$, and $\cos(0.1) \approx 0.9950042$.
• The polynomial is an approximation. It is closest near the center and usually drifts away farther out.
• Angles are in radians. The series for $\sin x$ and $\cos x$ hold only with $x$ in radians.
• MathBuddy can make mistakes. Double check important steps.

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