A friendly introduction to Taylor & Maclaurin series — for anyone about to take AP Calculus.
Here's a strange and wonderful fact: almost any curve can be rebuilt using nothing but
powers of x — x, x², x³, and so on. A "Taylor series" is the recipe for doing that. You start
with a rough guess and keep adding terms, and each term snaps the approximation a little closer
to the real thing. This page lets you watch it happen.
1 · Build it one term at a time
Pick a function, then hit Add a term and watch. Read the box under the
graph each time — it tells you the job the new term just did.
the real functionour approximationthe error (gap)where it stops working
Choose a function
The polynomial so far
2 · Moving the center: Maclaurin vs. Taylor
Here's something you may not have noticed: every approximation above was built around
one point — the center at x = 0 (the dark dot). Building a series at zero is so common it
gets its own name, the Maclaurin series. But the center doesn't have to be zero.
Drag the center slider below and watch what happens.
the real functionapproximationthe error (gap)center point
Choose a function
Center point
slide mex = 0
The polynomial so far
3 · The secret: each term matches one more thing
Why does adding terms work so neatly? Because of derivatives — the calculus idea
of "how fast something is changing." Each term is tuned so the approximation matches the real
function a little more deeply at the center point:
The first term matches the function's height. The next matches its slope
(that's the tangent line). The next matches its curvature — how the slope is bending. Keep
going and each term matches the next derivative, so the fake curve and the real one agree more and
more perfectly right at the center. The powers of x carry that agreement outward to the
neighborhood around it.
4 · Shortcuts: building new series from ones you know
Computing derivatives over and over is slow. Once you know a few core series, you can
reach a huge number of others with three moves — no new derivatives required. This is its own tested
skill, and it starts from one template.
The template everything builds on
The geometric series is the master key — memorize it cold:
① Substitute. Replace x with another expression to get a new series
instantly. Put −x² in for x:
1/(1+x²) = Σ(−x²)ⁿ = Σ(−1)ⁿx²ⁿ — no derivatives needed, just substitution.
② Integrate. Integrate a known series term by term to get another function's series.
Integrating 1/(1+x²) gives arctan x = Σ(−1)ⁿx²ⁿ⁺¹/(2n+1);
integrating 1/(1+x) gives ln(1+x) = Σ(−1)ⁿxⁿ⁺¹/(n+1).
③ Differentiate. Differentiate term by term to go the other way. Differentiating
1/(1−x) = Σxⁿ gives 1/(1−x)² = Σ n·xⁿ⁻¹.
The catch worth remembering: these moves keep the same radius of convergence
as the series you started from (the endpoints can change, but the radius doesn't). So if your template
works for |x| < 1, so do the series you build from it.
5 · Drill it — four ways to practice
Pick a mode and practice: read off a coefficient, write a full term, build a new
series by substitution, or judge where a series converges. Each one shows a full worked solution
that derives the answer, not just states it.
The one formula behind all of this
Every Maclaurin series is built from a single idea: the coefficient of xⁿ is
chosen so the polynomial's n-th derivative matches the function's at
x = 0. That gives the formula:
In words: take the n-th derivative of f, plug in
0, and divide by n!. That single rule produces every
coefficient — and it explains the patterns you'll see below:
Why factorials appear: dividing by n! every time is where the
1/n! in eˣ, sin, and cos comes from.
Why some powers vanish:sin x is odd, so its even
derivatives are 0 at the origin — every even power drops out. cos x is
even, so the odd powers drop out. A "0 coefficient" is normal, not a mistake.
Why signs alternate: the derivatives of sine and cosine cycle through
+, +, −, −, which flips the sign every other nonzero term.
Why substitution works: if you already know the series for f(x),
then f(x²) is just that series with x² in place of
x — no new derivatives needed. That shortcut is the whole trick.