Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — derive all: Foundations

This page assumes nothing. Every symbol the parent note (parent topic) throws at you is unpacked here, in an order where each idea leans only on the ones before it.

0. What is a function $f (x)$ ?

The picture. Draw a horizontal number line for inputs and a vertical one for outputs. For each input $x$ you get a height $f (x)$ ; joining all those heights traces a curve (its graph).

Figure — Maclaurin series of eˣ, sin x, cos x, ln(1+x), (1+x)ⁿ — derive all

1. Powers and $x^{n}$

The picture. As $n$ grows, $x^{n}$ for a small input like $x = 0.1$ shrinks dramatically: $0.1, 0.01, 0.001, \dots$ . This "high powers become tiny near zero" is exactly why later terms in a series barely matter close to $x = 0$ .

2. Slope, and the derivative $f^{'} (x)$

The picture. Zoom in on the curve until it looks like a straight line — that line's steepness (rise over run) is $f^{'} (x)$ . Positive slope = going up, negative = going down, zero = flat.

Higher derivatives $f^{''}$ , $f^{'''}$ , $f^{(n)}$

3. The factorial $n!$

The picture. Factorials explode: $1, 1, 2, 6, 24, 120, 720, \dots$ . Picture a staircase whose steps get impossibly tall very fast.

4. Summation notation $\sum$

Read it slowly: $\sum_{n = 0}^{\infty} \frac{x ^{n}}{n !} = n = 0 \frac{x ^{0}}{0 !} + n = 1 \frac{x ^{1}}{1 !} + n = 2 \frac{x ^{2}}{2 !} + \dots = 1 + x + \frac{x ^{2}}{2} + \dots$

5. Infinite sums that actually settle — convergence

The picture. Plot the running total (called a partial sum) after $1, 2, 3, \dots$ terms. If those dots settle toward a horizontal level, the series converges to that level. If they run off to infinity or jump around, it diverges.

6. The geometric series — our one pre-built engine

The picture. Each term is a fixed fraction of the one before, so the terms form a shrinking staircase whose total area is finite. Full derivation lives at Geometric series.

7. Integration (undoing the derivative)

8. Binomial coefficients $(k n)$

9. Odd and even functions (a free sanity check)

The picture. Even = reflection (like $x^{2}$ , a symmetric bowl). Odd = pinwheel (like $x^{3}$ , up on one side, down on the other). The power $x^{n}$ itself is even when $n$ is even, odd when $n$ is odd.

10. The imaginary unit $i$ (a peek ahead)

Prerequisite map

Equipment checklist

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

What does

f^{'} (x)

measure at a point?

The slope (steepness) of the curve there.

What does the superscript in

f^{(4)} (x)

mean?

Differentiate

f

four times — it is a counter, not a power.

Compute

4!

1 \cdot 2 \cdot 3 \cdot 4 = 24

What is

0!

and why?

0! = 1

; an empty product is the neutral number

1

What is

\frac{d ^{n}}{d x ^{n}} x^{n}

n!

— this is why every Maclaurin coefficient carries a

\frac{1}{n !}

Expand

n = 0 \sum 2 \frac{x ^{n}}{n !}

1 + x + \frac{x ^{2}}{2}

When does

1 + r + r^{2} + \dots

converge, and to what?

When

∣ r ∣ < 1

; it converges to

\frac{1}{1 - r}

Write

\frac{1}{1 + x}

as a geometric series.

1 - x + x^{2} - x^{3} + \dots

(ratio

r = - x

What does

\int x^{n} d x

give?

\frac{x ^{n + 1}}{n + 1} + C

Evaluate

(2 1/2)

\frac{\frac{1}{2} \cdot ( - \frac{1}{2} )}{2 !} = - \frac{1}{8}

sin x

odd or even, and what does that force?

Odd; only odd powers survive in its series.

What does "interval of convergence" tell you?

The exact set of

x

where the series equals the function.

Connections

Taylor series (general centre a) — the general recipe; Maclaurin is its $a = 0$ special case
Geometric series — the pre-built engine behind $ln (1 + x)$
Radius & interval of convergence — where each series is legally valid
Binomial theorem (integer n) — the terminating case of $(k n)$
Euler's formula — where the symbol $i$ eventually earns its place
L'Hôpital & limits via series — a payoff once the series exist