Taylor and Maclaurin expansions: polynomials, remainder and limits
The Taylor polynomial, Peano and Lagrange remainders, the expansions to know, computing limits with Taylor and applications to linearisation.
The tangent line approximates a function near a point. The Taylor polynomial does the same with a parabola, a cubic, a polynomial of any degree, and the approximation improves with every extra degree. It is the most powerful tool of first-year calculus for limits, and the one calculators use to compute sines and exponentials.
The Taylor polynomial
If 
is 
times differentiable at 
, its Taylor polynomial of order is
![\[ T_n(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{f''(x_0)}{2!}(x - x_0)^2 + \dots + \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n. \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-b9eb71d555995127e90b7ea588513865_l3.png "Rendered by QuickLaTeX.com")
It is the only polynomial of degree 
with the same value and the same first derivatives as at . With 
it is called the Maclaurin polynomial.
The remainder
The error 
can be written in two ways:
- Peano remainder:
, that is the error is negligible compared with 
as 
. This is what you need for limits; - Lagrange remainder:
for some 
between and 
. This is what you need to estimate the error numerically.
The symbol 
(“little o of g”) denotes a quantity that, divided by 
, tends to 0. Useful rules: 
, 
, 
if 
.
Expansions to know (x → 0)
 |  |
 |  |
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Expansions of composite functions come by substitution: 
, 
.
Computing limits with Taylor
Expand numerator and denominator up to the first term that does not cancel.
![\[ \lim_{x\to0}\frac{e^x - 1 - x}{x^2} = \lim_{x\to0}\frac{\frac{x^2}{2} + o(x^2)}{x^2} = \frac12. \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-b2e1d8d17f5e717ad36a49509c16462d_l3.png "Rendered by QuickLaTeX.com")
![\[ \lim_{x\to0}\frac{\cos x - e^{-x^2/2}}{x^4} = \lim_{x\to0}\frac{\left(1 - \frac{x^2}{2} + \frac{x^4}{24}\right) - \left(1 - \frac{x^2}{2} + \frac{x^4}{8}\right) + o(x^4)}{x^4} = \frac{1}{24} - \frac18 = -\frac{1}{12}. \]](https://www.electronics-engineering.com/wp-content/ql-cache/quicklatex.com-e77ce193e2058a95b8104e9f2067ec06_l3.png "Rendered by QuickLaTeX.com")
In the second example first-order equivalences give 
and l’Hôpital would need four derivatives: Taylor takes two lines. The practical rule is to push the expansion up to the order of the denominator.
Other uses
- Nature of stationary points: if
and the first non-zero derivative has order , for even the point is an extremum (a minimum if 
), for odd it is an inflection point. - Numerical computation:
, with a Lagrange error below 
. - Linearisation: in electronics the “small-signal” models of the diode and the transistor are the first-order expansion of the exponential characteristic around the operating point, and the second- and third-order terms are the harmonic distortion.
