Taylor Series

Taylor’s Theorem

If $f$ has derivatives up to order $(n+1)$ near $a$, then:

$f(a+h) = f(a) + hf'(a) + \frac{h^2}{2!}f''(a) + \cdots + \frac{h^n}{n!}f^{(n)}(a) + R_{n+1}$

Lagrange remainder: $R_{n+1} = \frac{h^{n+1}}{(n+1)!}f^{(n+1)}(\xi)$

where $\xi$ lies between $a$ and $a + h$.

Interpretation: We can approximate smooth functions by polynomials, with the remainder giving the error.

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Maclaurin Series

The special case of Taylor series with $a = 0$:

$f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + \cdots$

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Standard Power Series

These series should be memorised:

Function	Power Series	Convergence
$e^x$	$\sum_{k=0}^{\infty} \frac{x^k}{k!}$	All $x$
$\sin x$	$\sum_{k=0}^{\infty} \frac{(-1)^k x^{2k+1}}{(2k+1)!}$	All $x$
$\cos x$	$\sum_{k=0}^{\infty} \frac{(-1)^k x^{2k}}{(2k)!}$	All $x$
$\sinh x$	$\sum_{k=0}^{\infty} \frac{x^{2k+1}}{(2k+1)!}$	All $x$
$\cosh x$	$\sum_{k=0}^{\infty} \frac{x^{2k}}{(2k)!}$	All $x$
$\ln(1+x)$	$\sum_{k=1}^{\infty} \frac{(-1)^{k+1} x^k}{k}$	$-1 < x \leq 1$
$\ln(1-x)$	$-\sum_{k=1}^{\infty} \frac{x^k}{k}$	$-1 \leq x < 1$

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Binomial Expansion

For $|x| < 1$ and any $\alpha \in \mathbb{R}$:

$(1+x)^\alpha = 1 + \alpha x + \frac{\alpha(\alpha-1)}{2!}x^2 + \frac{\alpha(\alpha-1)(\alpha-2)}{3!}x^3 + \cdots$

Special cases:

• $\alpha = n$ (positive integer): terminates (binomial theorem)
• $\alpha = -1$: $\frac{1}{1+x} = 1 - x + x^2 - x^3 + \cdots$
• $\alpha = \frac{1}{2}$: $\sqrt{1+x} = 1 + \frac{x}{2} - \frac{x^2}{8} + \frac{x^3}{16} - \cdots$

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Manipulating Series

Addition: $\sum a_n x^n + \sum b_n x^n = \sum (a_n + b_n)x^n$

Multiplication: $(\sum a_n x^n)(\sum b_n x^n) = \sum c_n x^n$ where $c_n = \sum_{k=0}^{n} a_k b_{n-k}$

Composition: Substitute one series into another

Differentiation: $\frac{d}{dx}\sum a_n x^n = \sum n a_n x^{n-1}$ (within radius of convergence)

Integration: $\int \sum a_n x^n \,dx = \sum \frac{a_n x^{n+1}}{n+1}$ (within radius of convergence)

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Radius of Convergence

A power series $\sum a_n x^n$ converges absolutely for $|x| < R$ and diverges for $|x| > R$, where:

$R = \lim_{n \to \infty} \left|\frac{a_n}{a_{n+1}}\right| \quad \text{(ratio test)}$

At the endpoints $x = \pm R$, convergence must be checked separately.

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Example Expansions

Expand $\tan x$ about $x = 0$:

Since $\tan x$ is odd, write $y = \tan x$. Taking derivatives:

• $y(0) = 0$
• $y' = \sec^2 x$, so $y'(0) = 1$
• $y'' = 2\sec^2 x \tan x$, so $y''(0) = 0$
• $y''' = 2\sec^4 x + 4\sec^2 x\tan^2 x$, so $y'''(0) = 2$

Thus: $\tan x = x + \frac{x^3}{3} + O(x^5)$

Expand $(1+x)\ln(1+x)$:

Using $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$:

$(1+x)\ln(1+x) = x + \left(1 - \frac{1}{2}\right)x^2 + \left(\frac{1}{3} - \frac{1}{2}\right)x^3 + \cdots = x + \frac{x^2}{2} - \frac{x^3}{6} + \cdots$

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Why Taylor Series Matter

• Approximation: Truncating gives polynomial approximations
• Limits: Evaluate indeterminate forms (L’Hôpital’s rule basis)
• Numerical methods: Derive finite difference schemes
• Physics: Linearisation of nonlinear equations
• ODEs: Power series solutions

The Taylor series is the bridge between “nice” functions (exponential, trig) and polynomials, which we can compute with.