Kit Library / Mathematics / Calculus
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💡 Key Idea
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.
↳ The Taylor series encodes a function's entire local behavior in the derivatives at one point.
📖 Definition
Same formula, one key shortcut.
↳ Every Maclaurin series is a Taylor series; not every Taylor series is a Maclaurin series.
➗ Formula
These five cover most exam questions.
↳ Memorizing these five standard expansions lets you derive most others by substitution or differentiation.
⚙️ Process
Five steps, any function, any center.
↳ The pattern of derivatives at the center determines the entire series.
⚠️ Common Mistake
Missing n! ruins the whole expansion.
↳ The $n!$ factor is what makes the series converge to the correct function — never omit it.
⚠️ Common Mistake
Center a = 0? Then use Maclaurin.
↳ Match the center in the formula to the center in the problem — $(x-a)$, not $x$.
⚖️ Comparison
Two names, one formula, one difference.
↳ Maclaurin is just Taylor with the center pinned at zero.
🛠️ Application
Estimate sin(0.1) by hand in seconds.
↳ Taylor polynomials turn transcendental functions into simple arithmetic.
🎯 Exam Tip
Don't re-derive. Substitute and go.
↳ Substitution into known expansions is faster and safer than computing derivatives from scratch.
🧠 Memory Trick
Sin starts odd, cos starts even.
↳ Sine = odd powers, cosine = even powers, both with alternating signs.
🚫 Warning
A series may only work in a range.
↳ A Taylor series is only equal to the function inside its interval of convergence.
❓ Did You Know
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 $e^{i\theta} = \cos\theta + i\sin\theta$.
↳ Taylor series connect exponential and trigonometric functions through complex numbers.
Limits become algebra with series.
↳ Series expansion is often the fastest route through 0/0 limits.
Master these first, or it gets messy.
↳ Solid derivative skills are the single biggest prerequisite for Taylor series.
🌍 Real World
Your calculator, GPS, and games use them.
↳ Taylor series are the hidden engine behind numerical computation everywhere.
Taylor and Maclaurin series express a smooth function as an infinite sum of polynomial terms built from its derivatives at a single point. The Maclaurin series is simply the Taylor series centered at zero, and together t…
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