Taylor vs. Maclaurin Series

Taylor vs. Maclaurin series comes down to one fact: a Maclaurin series is a Taylor series centered at $0$. If the center is $a = 0$, it is Maclaurin. If the center is any other value, it is a Taylor series.

That sounds like a small naming change, but the center matters because a series is usually most useful near the point where it is built.

The Difference In One Formula

If a function has enough derivatives at $a$, its Taylor series about $a$ is

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

Set $a = 0$, and you get the Maclaurin series:

$$\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$

So the structure does not change. The center changes.

Why The Center Matters

The coefficients come from derivatives evaluated at the center. Change the center, and the numbers in the series usually change too.

A Maclaurin series is built to describe the function near $x = 0$. A Taylor series about $a = 2$ is built to describe the same function near $x = 2$. Both can be correct, but one may be much more practical for the value you care about.

You should also avoid a stronger claim than the problem allows. A Taylor or Maclaurin series is always designed as a local expansion. Whether it actually equals the function on an interval depends on the function and where the series converges.

Worked Example: $e^x$ At Two Different Centers

Take

$$f(x) = e^x$$

This is a good comparison because every derivative of $e^x$ is still $e^x$.

Maclaurin series at $a = 0$

At $a = 0$, every derivative value is $f^{(n)}(0) = 1$, so

$$e^x = \sum_{n=0}^{\infty}\frac{x^n}{n!}$$

The first few terms are

$$1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$$

Taylor series at $a = 1$

Now center the same function at $a = 1$. Then every derivative value at the center is $f^{(n)}(1) = e$, so

$$e^x = \sum_{n=0}^{\infty}\frac{e}{n!}(x-1)^n$$

The first few terms are

$$e + e(x-1) + \frac{e}{2!}(x-1)^2 + \cdots$$

The function stayed the same. Only the center changed. That is the whole difference between Taylor and Maclaurin series in one example.

When To Use Maclaurin Or Taylor

Use a Maclaurin series when $0$ is the natural reference point or when derivatives at $0$ are easy to compute.

Use a Taylor series around another value $a$ when you need a good local approximation near that value. For example, if you want to estimate behavior near $x = 3$, expanding around $a = 3$ is usually better than expanding around $0$.

Common Mistakes Students Make

Treating them as different ideas

They are not different theories. Maclaurin is one special case of Taylor.

Ignoring the center

Two series for the same function can both be valid, but the one centered near your target value is usually the more useful approximation.

Assuming the series always equals the function

That is not automatic. The answer depends on the function and the interval. The safe statement is that the series gives a local expansion around its center, and then you check convergence if the problem asks for more.

Where You See This In Calculus

Taylor and Maclaurin series show up when you approximate functions, study local behavior, solve differential equations, or replace a complicated expression with a polynomial that is easier to work with.

The recurring question is simple: which point makes the local model most useful?

Try A Similar Problem

Write the series for $\sin x$ twice: once at $a = 0$ and once at $a = \pi/4$. Comparing those two expansions is one of the fastest ways to make the difference stick.