Taylor and Maclaurin Series: quick revision, 15 bites

A Function Rebuilt From Its Derivatives

One point. Infinite derivatives. Whole function.

A Taylor series rebuilds a smooth function near a point $a$ using only the function's derivatives at that point.

Takeaway. The Taylor series encodes a function's entire local behavior in the derivatives at one point.

Taylor vs Maclaurin: What's the Difference?

Same formula, one key shortcut.

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

The Five Expansions You Must Memorize

These five cover most exam questions.

Takeaway. Memorizing these five standard expansions lets you derive most others by substitution or differentiation.

How to Build a Taylor Series Step by Step

Five steps, any function, any center.

Takeaway. The pattern of derivatives at the center determines the entire series.

Forgetting the Factorial Denominator

Missing n! ruins the whole expansion.

Takeaway. The $n!$ factor is what makes the series converge to the correct function — never omit it.

Using the Wrong Center

Center a = 0? Then use Maclaurin.

Takeaway. Match the center in the formula to the center in the problem — $(x-a)$, not $x$.

Taylor vs Maclaurin at a Glance

Two names, one formula, one difference.

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

Approximating Values Without a Calculator

Estimate sin(0.1) by hand in seconds.

Takeaway. Taylor polynomials turn transcendental functions into simple arithmetic.

Exam Pattern: Standard Expansions Save Time

Don't re-derive. Substitute and go.

Takeaway. Substitution into known expansions is faster and safer than computing derivatives from scratch.

Remember the Sine and Cosine Signs

Sin starts odd, cos starts even.

Takeaway. Sine = odd powers, cosine = even powers, both with alternating signs.

Convergence Isn't Automatic

A series may only work in a range.

Takeaway. A Taylor series is only equal to the function inside its interval of convergence.

Euler's Famous Formula Comes From Taylor

e^(iπ) + 1 = 0 — from series.

Substituting $x = i\theta$ into the Maclaurin series for $e^x$ and separating real and imaginary parts gives Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$.

Takeaway. Taylor series connect exponential and trigonometric functions through complex numbers.

Fast Limit Evaluation with Series

Limits become algebra with series.

Takeaway. Series expansion is often the fastest route through 0/0 limits.

What You Need Before Taylor Series

Master these first, or it gets messy.

Takeaway. Solid derivative skills are the single biggest prerequisite for Taylor series.

Where Taylor Series Power the Real World

Your calculator, GPS, and games use them.

Takeaway. Taylor series are the hidden engine behind numerical computation everywhere.