Mathematical Tools II: Taylor Series Expansion

IngéSUP - EN Section Mathematical Tools II: Taylor Series Expansion 2025-2026

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1. Relations of Comparison Between Functions

Let I be an open interval and let ( f, g : I \rightarrow \mathbb{R} ) and ( a \in I ) be such that ( g ) is not zero in any neighborhood of ( a ).

• The function ( f ) is called negligible compared to ( g ) around ( a ). We denote this as:

  • ( f(x) = o_a(g(x)) ) if ( \frac{f(x)}{g(x)} \xrightarrow{\, x \rightarrow a \;} 0 ).

• The functions ( f ) and ( g ) are called equivalent around ( a ). This is denoted as:

  • ( f(x) \sim_a g(x) ) if ( \frac{f(x)}{g(x)} \xrightarrow{\, x \rightarrow a \;} 1 ).

• These concepts can also be defined in the neighborhood of infinity:

  • ( f(x) = o\infty(g(x)) ) and ( f(x) \sim\infty g(x) ).

Definitions for General Cases

Let ( a \in \mathbb{R} \cup {\pm\infty} ); ( f ) and ( g ) be any two functions defined on an interval I containing ( a ):

• ( f(x) = a o(g(x)) ) if there exists ( \varepsilon : I \rightarrow \mathbb{R} ) such that ( \forall x \in I, f(x) = \varepsilon(x)g(x) ) and ( \varepsilon(x) \xrightarrow{\, x \rightarrow a \;} 0 ).

• ( f(x) \sim_a g(x) ) if there exists ( \varepsilon : I \rightarrow \mathbb{R} ) such that ( \forall x \in I, f(x) = (1 + \varepsilon(x))g(x) ) and ( \varepsilon(x) \xrightarrow{\, x \rightarrow a \;} 0 ).

Properties of Relations

• Reflexive: ( f \sim_a f )

• Symmetric: ( f \sima g \Leftrightarrow f - g =a o(g) \Leftrightarrow g - f =a o(f) \Leftrightarrow g \sima f )

• Transitive: If ( f \sima g ) and ( g \sima h ), then ( f \sim_a h ).

• Product: If ( f1 \sima g1 ) and ( f2 \sima g2 ), then ( f1f2 \sima g1g_2 ).

• Reciprocal: If ( f \sima g), then ( \frac{1}{f} \sima \frac{1}{g} ).

• Power: If ( f \sima g ) and ( \alpha \in \mathbb{R} ), then ( f^\alpha \sima g^\alpha ).

• Limit: If ( f \sima g ) and ( \lim{x \to a} g(x) = l ), then ( \lim_{x \to a} f(x) = l ).

Transforming Equivalences Around 0

• By letting ( x = a + h ): ( f(x) \sima g(x) \Leftrightarrow f(a + h) \sim0 g(a + h) ).

• Setting ( x = \frac{1}{t} ) leads to:

  • For ( +\infty ): ( f(x) \sim{+\infty} g(x) \Leftrightarrow f(\frac{1}{t}) \sim0 g(\frac{1}{t}) ).

  • For ( -\infty ): ( f(x) \sim{-\infty} g(x) \Leftrightarrow f(\frac{1}{t}) \sim0 g(\frac{1}{t}) ).

Remarks on Specific Functions

• ( \sin(x) \sim_0 x )

• ( \cos(x) - 1 \sim_0 -\frac{x^2}{2} )

• ( e^x - 1 \sim_0 x )

• ( \ln(x + 1) \sim_0 x )

• For all ( \alpha \in \mathbb{R}), ( (1 + x)^\alpha - 1 \sim_0 \alpha x ).

• Growth Comparisons:

  • For all ( r > 0 hinspace, ): ( \ln(x) = +\infty hinspace o(x^{r}) ) ( \Rightarrow ex = -\infty \thinspace o(|x|^{-r}) )

  • If ( f(x) = a o(x^n) ), then ( x^pf(x) = a o(x^{n+p}) ).

• Higher Powers: ( x^2 = 0 o(x) ) ( x^3 = 0 o(x^2) ) and for all ( n \geq p ), ( x^n = 0 o(x^p) ).

2. Taylor - Young Theorem

Let I be an open interval, and let ( f : I \rightarrow \mathbb{R} ) be a function of class ( C^n ) with ( a \in I ).

• Then, for all ( x \in I ):

  • ( f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + … + \frac{f^{(n)}(a)}{n!}(x - a)^n + (x - a)^n \varepsilon(x) ), where ( \varepsilon(x) \xrightarrow{\, x \rightarrow a \;} 0 ).

• This can be rewritten as:

  • ( f(x) = a f(a) + f'(a)(x - a) + \frac{f''(a)}{2!} (x - a)^2 + … + \frac{f^{(n)}(a)}{n!} (x - a)^n + o((x - a)^n) ).

• In the vicinity of 0, the Taylor-Young formula takes the form:

  • ( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + … + \frac{f^{(n)}(x)}{n!} x^n + x^n \varepsilon(x) ) where ( \varepsilon(x) \xrightarrow{\, 0 \rightarrow 0 \;} 0 ).

3. Taylor Series Expansion in the Neighborhood of a Point

Let I be an open interval, ( f : I \rightarrow \mathbb{R} ) with ( a \in I ) and ( n \in \mathbb{N} ).

• We say that ( f ) has a Taylor series expansion around ( a ) of order ( n ), denoted by ( TSE(f, a, n) ), if for all ( x \in I ):

  • ( f(x) = Pn(x - a) + (x - a)^n \varepsilon(x) ), where ( Pn ) is a polynomial of degree less than or equal to n and ( \varepsilon(x) \xrightarrow{\, x \rightarrow a \;} 0 ).

• The term ( P_n(x - a) ) is called the polynomial part, while ( (x - a)^n \varepsilon(x) ) is known as the remainder or error term.

Proposition

• If a function ( f ) has a Taylor series expansion around ( a ) of order n, then the expansion is unique.

Remarks

• If ( f ) has a Taylor series expansion around 0:

  • If ( f ) is even, the polynomial part contains only even degree terms.

  • If ( f ) is odd, the polynomial part contains only odd degree terms.

• A function ( f ) has a Taylor series around a point ( a ) of order n if and only if the function ( x \rightarrow f(x + a) ) has a Taylor series expansion around 0. Therefore, we often use the substitution ( h = x - a ) to simplify the problem around 0.

Additional Insights

• If a function ( f ) has a Taylor series expansion around ( x_0 ) of order ( n \geq 2 ):

  • The equation of the tangent to the graph of ( f ) at ( x0 ) is given by ( y = a0 + a1(x - x0) ).

  • The sign of ( an(x - x0)^n ) determines the relative (and local) positioning of the graph of ( f ) to the tangent line.

4. Taylor Series Expansions of Usual Functions Around 0

Some common Taylor series expansions of functions around 0 include:

• ( e^x = 1 + x + \frac{x^2}{2!} + … + \frac{x^n}{n!} + x^n \varepsilon(x) ).

• ( \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} + … + (-1)^n \frac{x^{2n+1}}{(2n + 1)!} + x^{2n+1} \varepsilon(x) ).

• ( \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} + … + (-1)^n \frac{x^{2n}}{(2n)!} + x^{2n} \varepsilon(x) ).

• For ( \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 + … + \frac{\alpha(\alpha - 1)…(\alpha - n + 1)}{n!} x^n + x^n \varepsilon(x) ).

5. Algebraic Operations on Taylor Series Expansions

5.1 Sum and Product

Let ( f ) and ( g ) be two functions each having a Taylor series expansion of order n at 0:

• ( TSE(f, 0, n): f(x) = a0 + a1x + … + anx^n + x^n \varepsilon1(x) )

• ( TSE(g, 0, n): g(x) = b0 + b1x + … + bnx^n + x^n \varepsilon2(x) )

• The function ( f + g ) has a Taylor series expansion of order n at 0:

  • ( TSE(f + g, 0, n): f + g(x) = (a0 + b0) + (a1 + b1)x + … + (an + bn)x^n + x^n \varepsilon(x) ).

• The function ( fg ) will have a Taylor series expansion of order n at 0:

  • ( TSE(fg, 0, n): fg(x) = Pn(x) + x^n \varepsilon(x) ) where ( Pn ) is the polynomial obtained by truncating the product ( (a0 + a1x + … + anx^n)(b0 + b1x + … + bnx^n) ) up to degree n.

5.2 Composition

Let ( f ) and ( g ) each having a Taylor series expansion of order n at 0 with ( f(0) = 0 ):

• If ( TSE(f, 0, n): f(x) = Pn(x) + x^n \varepsilon1(x), \; Pn(0) = 0 ) and ( TSE(g, 0, n): g(x) = Qn(x) + x^n \varepsilon_2(x) ):

• The function ( g \circ f ) has a Taylor series expansion of order n at 0:

  • ( TSE(g\circ f, 0, n): g\circ f(x) = Rn(x) + x^n \varepsilon(x) ), where ( Rn ) is the polynomial obtained by truncating ( Qn \circ Pn ) up to degree n.

5.3 Integration

Let I be an open interval, and let ( f : I \rightarrow \mathbb{R} ) be a function of class ( C^n ) that has a Taylor series expansion of order n at ( x_0 \in I):

• Then an anti-derivative ( F ) of ( f ) in I, has a Taylor series expansion of order n + 1 at ( x_0 ):

  • ( F(x) = F(x0) + a0(x - x0) + a1 \frac{(x - x0)^2}{2} + … + an \frac{(x - x0)^{n + 1}}{n + 1} + (x - x0)^{n + 1} \varepsilon_1(x) ).

6. Taylor Series Expansion at +∞: Asymptotic Taylor Series

Let ( I = ]\alpha, +\infty[ ), and let ( f : I \rightarrow \mathbb{R} ) with ( n \in \mathbb{N} ).

• We say that ( f ) has a Taylor series expansion around +∞ if there exist reals ( a0, a1,…, a_n ) such that:

  • ( f(x) = a0 + a1 x + a2 x^2 + … + an x^n + \frac{1}{x^n} \varepsilon \left( \frac{1}{x} \right) ) with ( \varepsilon \left( \frac{1}{x} \right) \xrightarrow{\, x \rightarrow +\infty \;} 0 ).

Remarks

• We can similarly define Taylor series expansions at -∞.

• A function ( f ) has a Taylor series expansion at +∞ of order n if and only if the function ( x \rightarrow f \left( \frac{1}{x} \right) ) has a Taylor series expansion at 0. This often requires substituting ( h = \frac{1}{x} ) to address the problem around 0.

• If the function ( x \rightarrow \frac{f(x)}{x} ) has a Taylor series expansion at +∞ of order ( n \geq 2 ):

  • ( f(x) ) can be expressed as: ( f(x) = a0x + a1 + a_n x^{n - 1} + \frac{1}{x^{n - 1}} \varepsilon \left( \frac{1}{x} \right) ).

  • There exists an asymptote at +∞ described by the equation ( y = a0x + a1 ).

  • The sign of ( a_n x^{n - 1} ) determines the relative (and local) positioning of the function ( f ) to the asymptote.