Calculus II · Unit 4B · lesson

Taylor Series Centered at a

Concept

Learning objectives

construct Taylor series about a nonzero center and interpret the variable $x-a$ .

Taylor Series Centered at a

Explanation

The center is the point where information is measured

A Taylor series centered at $a$ uses powers of $x-a$ , because approximation quality is organized by distance from $a$ . The coefficients encode derivative data at that center. Changing centers changes every coefficient even though the underlying function remains the same.

Nonzero centers are useful when approximating near a specific input or when a Maclaurin series converges too slowly for the desired calculation. Translating the variable with $h=x-a$ often makes the pattern clearer. The final series should state both its center and its convergence interval.

Bridge

Move the center to where the approximation is needed

A Taylor series centered at $a$ uses powers of $x-a$ , so its accuracy is organized by distance from $a$ . Choosing the center near the target input often makes the expansion variable small and the approximation efficient.

The center also changes which derivative values are convenient. A logarithm near $1$ , for example, is naturally expanded in powers of $x-1$ , while trigonometric functions near zero favor Maclaurin series. Center choice is a modeling decision, not merely notation.

**Taylor accuracy is local to the chosen center**
The same curve is shown with two local polynomial approximations, each closely matching near its own center and separating farther away.
Taylor accuracy is local to the chosen center. The same function with Taylor models centered at two different points.

Concept

Taylor series

The Taylor series of $f$ centered at $a$ is

\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n.

Guided walkthrough

Expand $\ln x$ around $1$

For $f(x)=\ln x$ ,

f(1)=0,\quad f'(1)=1,\quad f''(1)=-1,\quad f^{(3)}(1)=2.

Thus

\ln x=(x-1)-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}-\cdots,

for the appropriate interval around $1$ .

Worked example

Approximate ln(1.1) from a center at 1

Let $f(x)=\ln x$ and center at $a=1$ . Since

f^{(n)}(1)=(-1)^{n-1}(n-1)!\quad(n\ge1),

we obtain

\ln x=(x-1)-\frac{(x-1)^2}{2}+\frac{(x-1)^3}{3}-\cdots.

At $x=1.1$ , using three terms gives

0.1-\frac{0.1^2}{2}+\frac{0.1^3}{3}=0.095333\ldots,

close to the true value because $x-1=0.1$ is small.

Common mistake

Keep the expansion variable visible

When centered at $a$ , write and manipulate powers of $x-a$ . Expanding them too early often hides the center and creates coefficient errors.

Interactive checku4b-taylor_series_centered_at_a-01

What is the center of a series written in powers of $(x-1)$ ?

Your work stays on this device. No account or AI grader is used.

Show hint

The center makes the power factor zero.

Attempt once to unlock the solution

Submit an answer first. The hint is available now.

Exercise

Find the quadratic Taylor polynomial for $e^x$ at $a=1$ .

Exercise

Rewrite the geometric series around $x=2$ for $1/(3-x)$ .

Exercise

Explain why powers of $x-a$ measure distance from the center.

Exercise

Compare Maclaurin and Taylor terminology.

After the explanation

Use the section idea

Reading lens

Match value and derivatives at one center, then separate the polynomial approximation from the infinite-series convergence claim.

Mental model

Taylor coefficients encode local derivative data as a polynomial of increasing degree.

Decision

Choose the center, compute the derivative pattern, divide by factorials, and state whether you need a polynomial or an infinite series.

Common trap

Assuming every smooth-looking function equals its Taylor series everywhere.

Check yourself

Can you verify the first coefficients directly from derivatives at the center?

Practice this skill

Practice this skillTaylor Series Worksheet with Answers and Error Practice

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.