Taylor and Maclaurin Series: quick revision, 16 bites

One Point, Infinite Polynomial

A function's DNA lives at one point.

A Taylor series rebuilds an entire function from its derivatives at a **single point** $a$ — turning complicated functions into infinite polynomials.

Takeaway. Every coefficient of the series is determined by one derivative value at the center.

Taylor vs Maclaurin: The Only Difference

Maclaurin is Taylor with a=0.

Takeaway. Every Maclaurin series is a Taylor series; not every Taylor series is a Maclaurin series.

The General Term Formula

Memorize this one formula, unlock all series.

Takeaway. Each term pairs one derivative with one factorial and one power of (x − a).

Maclaurin Series of e^x

e^x is its own derivative — see why.

Since every derivative of $e^x$ equals $e^x$, and $e^0 = 1$, all coefficients become $1/n!$.

Takeaway. e^x has the cleanest Maclaurin series because all derivatives equal 1 at x = 0.

Maclaurin Series of sin x and cos x

Odd powers for sin, even for cos.

Takeaway. Symmetry of the function shows up as which powers survive in its series.

How to Build a Taylor Series

Five steps from function to series.

Takeaway. Derivatives at the center are the raw material; factorials and powers assemble them.

The Remainder Term

Truncation error has a name.

The series equals the function only when $R_n \to 0$ as $n \to \infty$.

Takeaway. Convergence isn't automatic — the remainder must vanish.

Taylor vs Maclaurin at a Glance

Same idea, different center.

Takeaway. Maclaurin is just Taylor with the center pinned at zero.

Forgetting the Factorial

Missing n! ruins the whole series.

Takeaway. The factorial is not optional — it's baked into the definition.

Using x Instead of (x − a)

Center a is not always zero.

Takeaway. The power of each term must mirror the center point.

Why Calculators Use Series

Your calculator never 'knows' sin x.

Calculators and computers evaluate $\sin x$, $e^x$, and $\ln x$ by summing a few Taylor terms — not by looking up a table.

Takeaway. Taylor series turn 'impossible' functions into simple arithmetic.

Linear and Quadratic Approximations

First two terms solve most problems.

Takeaway. Truncating the Taylor series gives the best polynomial approximation of that degree.

Remember the Coefficient Pattern

Derivative over factorial, times power.

Chant it: **derivative over factorial, times the power**. That's the whole Taylor coefficient in one line.

Takeaway. f^(n)(a) / n! · (x − a)^n — say it, don't derive it.

Standard Series to Memorize

Five series cover 90% of exams.

Takeaway. Recognizing a standard series saves you from re-deriving it under time pressure.

Convergence Isn't Guaranteed Everywhere

Series can fail far from center.

Takeaway. A Taylor series represents its function only within its interval of convergence.

Euler's Beautiful Identity from Taylor

Plug iπ into e^x and see magic.

Substituting $x = i\pi$ into the Maclaurin series of $e^x$ yields $e^{i\pi} = -1$ — Euler's identity, one of the most celebrated equations in mathematics.

Takeaway. Taylor series connect exponentials, trigonometry, and complex numbers.