Taylor Series
Learn Taylor and Maclaurin series, memorise the standard expansions, and use them for small-parameter approximations in physics.
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The core idea
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Taylor series let you approximate a function near a point using a polynomial. In physics, they are the standard way to get small-angle and small-parameter approximations.
1) Taylor series about x = a
If f is sufficiently differentiable near a, then:
f(x) = ∑ₙ₌₀^∞(f⁽ⁿ⁾(a)/n!)(x-a)ⁿ.
The first few terms are:
f(x) = f(a) + f'(a)(x-a) + (f''(a)/2!)(x-a)² + (f⁽³⁾(a)/3!)(x-a)³ + …
2) Maclaurin series (special case: a = 0)
f(x) = ∑ₙ₌₀^∞(f⁽ⁿ⁾(0)/n!)xⁿ.
Practical approximation rule
If you truncate the series after a few terms, the first neglected term usually gives the order of the error (as long as x is in the convergence region).
3) Standard series you should memorise
Exponential
e^x = 1 + x + x²/2! + x³/3! + …
Sine and cosine
sin x = x-x³/3! + x⁵/5!-…
cos x = 1-x²/2! + x⁴/4!-…
Logarithm (for |x| < 1)
ln(1 + x) = x-x²/2 + x³/3-x⁴/4 + …
Binomial series (for |x| < 1)
For any real n:
(1 + x)ⁿ = 1 + nx + ((n(n-1))/2!)x² + ((n(n-1)(n-2))/3!)x³ + …
4) High-frequency approximations (physics favourites)
These come directly from the standard series:
- Small angle: sin x ≈ x and cos x ≈ 1-x²/2 for |x|≪ 1
- Small parameter: (1 + x)ⁿ ≈ 1 + nx for |x|≪ 1
- Log: ln(1 + x) ≈ x for |x|≪ 1
5) Worked examples
A. Approximate square root of (1 + x) to O(x²)
Use the binomial series with n = 1/2:
(1 + x)^(1/2) = 1 + (1/2)x + (((1/2)(-1/2))/2)x² + … = 1 + x/2-x²/8 + O(x³).
B. Linearise ln(1 + x) for small x
From the logarithm series:
ln(1 + x) = x-x²/2 + O(x³) ≈ x (|x|≪ 1).
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Common Taylor Series: