Foundations — Taylor series — derivation from power series

Before you can read the parent note Taylor series — derivation from power series, you must own the ten small ideas below. We go strictly in order: nothing appears before it is defined and drawn.

1. Function $f(x)$ — a height machine

The picture. Put $x$ on the horizontal axis. The output $f(x)$ is a height above (or below) that point. Sweep $x$ left-to-right and the heights trace a curve.

Why the topic needs it. Taylor series is about copying a curve. The thing we copy is exactly this height machine $f(x)$.

2. The point $a$ — the centre we build around

The picture. Mark a single dot on the $x$-axis at $x=a$. Everything we do happens near that dot.

Why. A Taylor series is only promised to be accurate near one place. That place is $a$.

3. Displacement $(x-a)$ — "how far from the centre"

The picture. Stand at $a$. If $x$ is to the right, $(x-a)$ is a small positive step; to the left, it is negative; and $(x-a)=0$ exactly at the centre.

4. Powers and $x^n$ — repeated multiplication

Why. Our copy is built from the pieces $(x-a{)}^0,(x-a{)}^1,(x-a{)}^2,\dots$ — the flat piece, the sloped piece, the curved piece, ... . Each power $n$ is one "shape ingredient".

5. Coefficients $c_n$ — how much of each ingredient

The picture. Think of a mixing desk. $c_0$ sets the base height, $c_1$ sets the tilt, $c_2$ sets the bend. Turning each dial changes how much of that ingredient goes in.

6. Subscript $n$ and the index — counting the terms

Why. We need to talk about "the general $n$-th term" all at once instead of writing infinitely many lines.

7. Sigma notation $\sum$ — a compact "add them all up"

The picture. A conveyor belt: the counter $n$ clicks $0,1,2,\dots$ and each click drops one term $T_n$ into a running total.

Why. The Taylor series is literally $\sum _{n=0}^{\infty }{c}_n(x-a{)}^n$. Without $\sum$ we could not write an infinite sum on one line.

8. Derivative $f^{\prime }(x)$ — instantaneous steepness

The picture. Zoom into the curve at a point until it looks straight. $f^{\prime }(x)$ is the steepness of that little straight segment — rise over run of the tangent line.

9. Higher derivatives $f^{\prime \prime }$, $f^{\prime \prime \prime }$, $f^{(n)}$ — slope of the slope

Why. In the parent derivation, evaluating $f^{(n)}(a)$ is what pins down the $n$-th coefficient.

10. Factorial $n!$ — the multiply-down staircase

The picture. A descending staircase of factors: start at $n$ and step down to $1$, multiplying every step.

11. Evaluating "at $x=a$" — the freeze trick

How these feed the topic

Equipment checklist

Test yourself — cover the right side and answer out loud.

Connections

• Parent: Taylor series — derivation
• Linear approximation & differentials (matching only height + slope)
• Geometric series (simplest power series: all $c_n=1$)
• Power series — radius & interval of convergence (when the infinite sum is even allowed)
• Maclaurin series — common expansions (the $a=0$ case you meet first)