Taylor Series Calculator

Build a Taylor or Maclaurin polynomial from a function, expansion center, and chosen degree.

Drag & drop or upload an image or PDF

Enter a request, such as degree 3 Taylor polynomial for ln(x) centered at 1

Try an example

Turn local derivative data into a polynomial

A Taylor polynomial matches a function and a chosen number of its derivatives at a center a. Near that center, the polynomial can be much easier to evaluate, differentiate, or integrate than the original function.

A Maclaurin polynomial is simply the case a = 0. The degree controls how many terms are kept. More terms often improve a nearby approximation, but accuracy also depends on the function, the center, and how far the target lies from that center.

How to use the taylor series calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

Taylor polynomial of degree n

Evaluate each derivative at the center, divide by the matching factorial, and multiply by the corresponding power of x - a.

T_{n} (x) = k = 0 \sum n \frac{f ^{(k)} ( a )}{k !} (x - a)^{k}

A reliable way to work through it

Record center and degree

The center determines the powers of x - a, while the degree determines the last derivative and term needed.

Build the coefficient table

Evaluate f(a), f′(a), and successive derivatives at a, then divide each value by its factorial.

State the approximation limits

Keep the remainder or local nature of the polynomial clear, especially when evaluating far from the center.

Worked example

Find the degree-3 Taylor polynomial for ln(x) at a = 1

f (1) = 0, f^{'} (1) = 1

Evaluate the function and first derivative at the center.

f^{''} (1) = - 1, f^{'''} (1) = 2

Continue through the third derivative.

T_{3} (x) = 0 + (x - 1) - \frac{( x - 1 ) ^{2}}{2} + \frac{2 ( x - 1 ) ^{3}}{6}

Substitute the values into the Taylor formula.

T_{3} (x) = (x - 1) - \frac{( x - 1 ) ^{2}}{2} + \frac{( x - 1 ) ^{3}}{3}

Simplify the cubic coefficient.

The degree-3 Taylor polynomial is (x - 1) - (x - 1)²/2 + (x - 1)³/3.

Common mistakes to check

Using powers of x at a nonzero center

A Taylor polynomial centered at a uses powers of x - a. Powers of x alone are for a Maclaurin center.

Forgetting the factorial

The kth derivative value is divided by k!, which changes every coefficient after the first derivative.

Treating an approximation as an identity

A finite Taylor polynomial is a local approximation, not the original function everywhere.

Derivative CalculatorDifferentiate a function, identify each rule, and simplify the result one step at a time.Limit CalculatorStudy what a function approaches from the left, right, or at infinity, even when direct substitution fails.Series Convergence CalculatorChoose a valid convergence test, check its conditions, and classify an infinite series.Integral CalculatorFind an indefinite integral, follow the chosen method, and keep the constant of integration.

Questions students ask

How are Taylor and Maclaurin series different?

A Maclaurin series is a Taylor series centered at zero.

How is a Taylor polynomial different from a Taylor series?

The polynomial stops at a chosen degree. The series continues infinitely.

Which center should I choose?

Choose a center near the point of interest where the function and needed derivatives are defined.

Does a Taylor series always equal its function?

No. Equality requires the series to converge to that function, not only that all derivatives exist.