In one line
Taylor's series writes \(f(x)\) as \(f(a)+(x-a)f'(a)+\dfrac{(x-a)^2}{2!}f''(a)+\dots\); Maclaurin's series is the special case \(a=0\).
Explained simply
A calculator cannot "know" \(\sin x\) or \(e^x\) directly; it only adds and multiplies. Series expansions turn these functions into polynomials that a machine can compute. It is like describing a curve near a point using its height, its slope, its bending, and so on: the more terms you use, the better the copy.
Technically, if \(f\) has derivatives of all orders at \(a\) (and the series converges), \(f(x)=\displaystyle\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\). With \(a=0\) this is Maclaurin's series. The module title mentions "Expansion of Function", and this is the standard tool for it; it also uses the successive derivatives from Module 4.
How it works
- Choose the point \(a\) (0 for Maclaurin).
- Find \(f(a),f'(a),f''(a),f'''(a),\dots\).
- Substitute into the series and simplify the coefficients.
- For products like \(e^x\sin x\), multiplying known series is often faster than differentiating.
Key formula
Taylor: \(f(x)=f(a)+(x-a)f'(a)+\dfrac{(x-a)^2}{2!}f''(a)+\dfrac{(x-a)^3}{3!}f'''(a)+\dots\)
Standard Maclaurin series: \(e^x=1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dots\); \(\sin x=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dots\); \(\cos x=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dots\); \(\log(1+x)=x-\dfrac{x^2}2+\dfrac{x^3}3-\dots\) for \(-1<x\le1\).
Quick example
(a) Expand \(f(x)=2x^3+7x^2+x-1\) in powers of \((x-2)\).
\(f(2)=45\), \(f'(x)=6x^2+14x+1\Rightarrow f'(2)=53\), \(f''(x)=12x+14\Rightarrow f''(2)=38\), \(f'''=12\).
\(f(x)=45+53(x-2)+19(x-2)^2+2(x-2)^3\). Check at \(x=3\): \(45+53+19+2=119=f(3)\).
(b) \(e^x\sin x=\left(1+x+\frac{x^2}2+\dots\right)\left(x-\frac{x^3}6+\dots\right)=x+x^2+\frac{x^3}3+\dots\)
Common mistakes
- Forgetting the factorials in the denominators.
- Using \(x^n\) instead of \((x-a)^n\) in a Taylor series about \(a\ne0\).
- Using \(\log(1+x)\) series for \(|x|>1\), where it does not converge.
How MU asks it
Note: the module title includes "Expansion of Function", but the detailed syllabus lists only numerical methods, so check with your teacher how much is examinable. If asked, typical 3–5-mark parts are "Expand \(\tan x\) in powers of \(x\) up to \(x^5\)" or "Expand \(\sqrt{x}\) in powers of \((x-4)\)". Show each derivative value at \(a\), substitute into the series, and simplify the coefficients.