Taylor & Maclaurin · Euler’s identity

Extra reference — series expansions, σ notation, and a video proof of \(e^{i\pi}+1=0\)

Maclaurin series (σ form) Euler’s identity via Taylor series More videos (add later)

§1 · Maclaurin expansions

Matches Calculus → Formulae → §3. This page adds σ notation and the YouTube proof in one place.

Standard series (|x| small enough for convergence)

Function	Series	Notes
\(e^x\)	\(\displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots\)	All real \(x\)
\(\sin x\)	\(\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots\)	All real \(x\)
\(\cos x\)	\(\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\)	All real \(x\)
\(\ln(1+x)\)	\(\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\)	\(-1 < x \le 1\)
\((1+x)^\alpha\)	\(\displaystyle\sum_{n=0}^{\infty}\binom{\alpha}{n}x^n\)	\(\|x\| < 1\)
\(\sinh x\)	\(\displaystyle\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}\)	Odd terms only
\(\cosh x\)	\(\displaystyle\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}\)	Even terms only

Taylor at \(x=a\)

\[f(x) = \sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k + R_n, \qquad R_n = \frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}.\]

Maclaurin = Taylor at \(a=0\). Coefficient of \(x^n\): \(f^{(n)}(0)/n!\).

§2 · Euler’s identity (\(e^{i\pi}+1=0\))

Videos

1 · Taylor series proof — plug Maclaurin series for \(e^x\), \(\cos x\), \(\sin x\); compare real and imaginary parts → \(e^{ix}=\cos x+i\sin x\), then \(x=\pi\).

▶ Taylor proof

2 · Intuition (Mathologer) — e to the pi i for dummies: geometric / conceptual path to \(e^{i\pi}=-1\) without getting lost in symbols.

▶ Mathologer — \(e^{\pi i}\)

Steps (Taylor proof — video 1)

\(e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots\)
Replace \(x\) by \(ix\): \(e^{ix} = 1 + ix - \dfrac{x^2}{2!} - i\dfrac{x^3}{3!} + \dfrac{x^4}{4!} + \cdots\)
Group real / imaginary: \(\mathrm{Re}(e^{ix}) = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \cdots = \cos x\)
\(\mathrm{Im}(e^{ix}) = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \cdots = \sin x\)
Euler’s formula: \(e^{ix} = \cos x + i\sin x\)
At \(x=\pi\): \(e^{i\pi} = \cos\pi + i\sin\pi = -1\) ⇒ \(e^{i\pi}+1=0\)

Useful corollaries (optional / complex analysis)

\(e^{x+iy} = e^x(\cos y + i\sin y)\)
\(\cos x = \dfrac{e^{ix}+e^{-ix}}{2}\), \(\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}\)
De Moivre: \((\cos\theta+i\sin\theta)^n = \cos n\theta + i\sin n\theta\)

§3 · More videos

Add links here as you collect them (same format as §2). Chapter formula sheets stay in each folder’s 00_Formulae_and_Theorems.html.