The Taylor-Young's formula with integral remainder

Let be $ f : x \longmapsto f(x) $ a function of class $ \mathcal{C}^n $ and $ f^{(n)} $ its $ n $-th derivative.

The Taylor-Young's formula

The Taylor-Young's formula tells us that any function $ f_{n,a} $, centered at $ x = a $, can be written as a Taylor series $ (TS_n(a)) $ with a remainder $R_n$, such as:

$$ f_{n,a}(x) = T_{n,a}(x) + R_{n,a}(x) \qquad (1) $$

Here is the decomposed form:

$$ \forall n \in \mathbb{N}, \enspace \forall f \in \hspace{0.05em} \mathcal{C}^n[D_f, \mathbb{R}], \enspace \forall (x, a) \in D_f^2, $$

$$ f_{n,a}(x) = f(a) + f'(a)(x - a) + \frac{f^{(2)}(a)}{2!}(x - a)^2 \hspace{0.2em} + \hspace{0.2em} ... \hspace{0.2em} + \hspace{0.2em} \frac{f^{(n)}(a)}{n!}(x - a)^n + \int_a^x \frac{f^{(n+1)}(t)}{n!}(x-t)^n \hspace{0.2em} dt $$

So,

$$ f_{n,a}(x) = \hspace{0.2em} \underbrace{ \sum_{k=0}^n \hspace{0.2em} \Biggl[ \frac{f^{(k)}(a)}{k!}(x - a)^k \Biggr] } _\text{regular part} \hspace{0.2em} + \hspace{0.2em} \underbrace{ \int_a^x \frac{f^{(n+1)}(t)}{n!}(x-t)^n \hspace{0.2em} dt } _\text{remainder } $$

Furthermore, setting down $ (x = a + h) $, we do obtain a new form of this formula:

$$ f_{n,a}(a + h) = f(a) + f'(a)h + \frac{f^{(2)}(a)}{2!}h^2 \hspace{0.2em} + \hspace{0.2em} ... \hspace{0.2em} + \hspace{0.2em} \frac{f^{(n)}(a)}{n!}h^n + \int_a^{a+h} \frac{f^{(n+1)}(t)}{n!}(a + h - t)^n \hspace{0.2em} dt $$

So,

$$ f_{n,a}(a + h) = \hspace{0.2em} \underbrace{ \sum_{k=0}^n \hspace{0.2em} \Biggl[ \frac{f^{(k)}(a)}{k!}h^k \Biggr] } _\text{regular part} \hspace{0.2em} + \hspace{0.2em} \underbrace{ \int_a^{a+h} \frac{f^{(n+1)}(t)}{n!}(a + h - t)^n \hspace{0.2em} dt } _\text{remainder } $$

Recap of the main Taylor series

Demonstration

The Taylor-Young's formula

Starting from the main equation of the fundamental theorem of calculus:

$$ f(x) = f(a) + \int_a^x f'(t)dt $$

So,

$$ f(x) = f(a) - \int_a^x -f'(t)dt $$

Performing an integration by parts, with a wise choice for $ u $ and $ v' $, we do have:

$$ \Biggl \{ \begin{align*} u(t) = f'(t) \\ v'(t) = -dt \end{align*} $$

$$ \Biggl \{ \begin{align*} u'(t) = f^{(2)}(t)dt \\ v(t) = x-t \end{align*} $$

$$ f(x) = f(a) - \Biggl( \Bigl[f'(t)(x-t)\Bigr]_a^x - \int_a^x f^{(2)}(t)(x-t) \hspace{0.2em} dt \Biggr) $$

$$ f(x) = f(a) + f'(a)(x-a) + \int_a^x f^{(2)}(t)(x-t)dt $$

Then we do it again with:

$$ \Biggl \{ \begin{align*} u(t) = f^{(2)}(t) \\ v'(t) = (x-t)dt \end{align*} $$

$$ \Biggl \{ \begin{align*} u'(t) = f^{(3)}(t)dt \\ v(t) = -\frac{(x-t)^2}{2} \end{align*} $$

$$ f(x) = f(a) + f'(a)(x-a) + \Biggl[ -\frac{f^{(2)}(t)}{2} (x-t)^2 \Biggr]_a^x + \int_a^x \frac{f^{(3)}(t)}{2}(x-t)^2dt $$

$$ f(x) = f(a) + f'(a)(x-a) + \frac{f^{(2)}(a)}{2} (x-a)^2 + \int_a^x \frac{f^{(3)}(t)}{2}(x-t)^2dt$$

And so on...

$$ f(x) = f(a) + f'(a)(x-a) + \frac{f^{(2)}(a)}{2} (x-a)^2 + \Biggl[ \frac{f^{(3)}(t)}{6} (x-t)^3 \Biggr]_a^x - \int_a^x \frac{f^{(4)}(t)}{6}(x-t)^3dt $$

$$ f(x) = f(a) + f'(a)(x-a) + \frac{f^{(2)}(x)}{2} (x-a)^2 + \frac{f^{(3)}(a)}{6} (x-a)^3 + \int_a^x \frac{f^{(4)}(t)}{6}(x-t)^3dt $$

Thus, the function$ f $ accept a Taylor series with integral remainder $ R_{n,a}(x) $ and this function is worth:

$$ \forall n \in \mathbb{N}, \enspace \forall f \in \hspace{0.05em} \mathcal{C}^n[D_f, \mathbb{R}], \enspace \forall (x, a) \in D_f^2, $$

So,

Furthermore, setting down $ (x = a + h) $, we do obtain a new form of this formula:

So,

Recap of the main Taylor series

Another notation used to characterize the remainder of an Taylor series is the Landau notation $o(x^n)$.

If a function $ f(x) $ is negligible compared to another function $ g(x) $ near a certain point $ a $, we can write it as:

$$f(x) \underset{x \to a}{=} o \bigl(g(x)\bigr)$$

It means that:

$$ \exists \varepsilon (x), \enspace f(x) \underset{x \to a}{=} \varepsilon (x) g(x) \qquad (with \enspace lim_{x \to a} \enspace \varepsilon (x) = 0)$$

In our specific case, we study Taylor series at the neighbourhood of $( a = 0 ) $, so:

$$ \exists \varepsilon (x), \enspace f(x) \underset{x \to 0}{=} \varepsilon (x) g(x) \qquad (with \enspace lim_{x \to 0} \enspace \varepsilon (x) = 0)$$

$$ condition $$	$$ function $$	$$ Taylor \ series \ at \ 0 \ : T_n(0) $$	$$ \equiv T_n(0) $$
$$ \forall x \in \mathbb{R}, \ \forall \alpha \in \hspace{0.05em} \mathbb{N}^*, $$	$$ (1+x)^{\alpha}$$ $$ (Newton's \ binomial) $$	$$ 1 + \alpha x + \binom{\alpha}{2}x^2 + \binom{\alpha}{3}x^3 \ ... \ + \binom{\alpha}{\alpha}x^{\alpha} + o(x^{\alpha})$$	$$ \sum_{p = 0}^{\alpha} \binom{\alpha}{p} x^k + o(x^{\alpha}) $$
$$ \forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \bigl\{ -1 \bigr\} \Bigr], $$	$$ \frac{1}{1+x}$$	$$ 1 - x + x^2 - x^3 + \ ... \ + (-1)^n x^n + o(x^n)$$	$$ \sum_{k=0}^n (-1)^k x^k + o(x^n) $$
$$ \forall x \in \Bigl[ \mathbb{R} \hspace{0.2em} \backslash \bigl\{ 1 \bigr\} \Bigr], $$	$$ \frac{1}{1-x}$$	$$ 1 + x + x^2 + x^3 + \ ... \ + x^n + o(x^n)$$	$$ \sum_{k=0}^n x^k + o(x^n) $$
$$ \forall x \in [-1, \hspace{0.1em} + \infty[, $$	$$ \sqrt{1+x}$$	$$ 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{4}x^3 - \frac{15}{16}x^4 \ ... \ + o(x^{4})$$	$$ $$
$$ \forall x \in \hspace{0.05em} ]-1, \hspace{0.1em} + \infty[, $$	$$ 1 \over \sqrt{1+x}$$	$$ 1 - \frac{1}{2}x + \frac{3}{4}x^2 - \frac{15}{8}x^3 + \frac{105}{16}x^4 \ ... \ + o(x^{4})$$	$$ $$
$$ \forall x \in \mathbb{R}, $$	$$ e^x $$	$$ 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ ... \ + \frac{x^n}{n!} + o(x^n)$$	$$ \sum_{k=0}^n \frac{x^k}{k!} + o(x^n) $$
$$ \forall x \in \hspace{0.05em} ]-1, \hspace{0.1em} + \infty[, $$	$$ ln(1+x) $$	$$ x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ ... \ + \frac{ (-1)^{n-1} }{n} x^n + o(x^{n})$$	$$ \sum_{k=1}^n \frac{ (-1)^{k-1} }{k} x^k + o(x^{n}) $$
$$ \forall x \in \hspace{0.05em} ]1, \hspace{0.1em} + \infty[, $$	$$ ln(1-x) $$	$$ -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \ ... \ - \frac{x^n}{n} + o(x^{n})$$	$$ \sum_{k=1}^n -\frac{ x^k }{k} + o(x^{n}) $$
$$ \forall x \in \mathbb{R}, $$	$$ sin(x) $$	$$ x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \ ... \ + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + o(x^{2n+2})$$	$$ \sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{(2k+1)!}+ o(x^{2n+2}) $$
$$ \forall x \in \mathbb{R}, $$	$$ cos(x) $$	$$ 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ ... \ + (-1)^{n} \frac{x^{2n}}{(2n)!} + o(x^{2n+1})$$	$$ \sum_{k=0}^n (-1)^{k} \frac{x^{2k}}{(2k)!} + o(x^{2n+1}) $$
$$\forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \Bigl\{ \frac{\pi}{2} + k\pi \Bigr\} \biggr],$$	$$ tan(x) $$	$$ x + \frac{1}{3}x^3 + \frac{2}{15}x^4 + \frac{17}{315}x^6 + \ ... \ + o(x^{6})$$	$$ $$
$$ \forall x \in \mathbb{R}, $$	$$ arctan(x) $$	$$ x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} \ ... \ + (-1)^n \frac{x^{2n+1}}{(2n+1)} + o(x^{2n+2})$$	$$ \sum_{k=0}^n (-1)^k \frac{x^{2k+1}}{(2k+1)}+ o(x^{2n+2}) $$
$$ \forall x \in \mathbb{R}, $$	$$ sinh(x) $$	$$ x + \frac{x^3}{3!} + \frac{x^5}{5!} + \frac{x^7}{7!} \ ... \ + \frac{x^{2n+1}}{(2n+1)!} + o(x^{2n+2})$$	$$ \sum_{k=0}^n \frac{x^{2k+1}}{(2k+1)!}+ o(x^{2n+2}) $$
$$ \forall x \in \mathbb{R}, $$	$$ cosh(x) $$	$$ 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \ ... \ + \frac{x^{2n}}{(2n)!} + o(x^{2n+1})$$	$$ \sum_{k=0}^n \frac{x^{2k}}{(2k)!} + o(x^{2n+1}) $$
$$ \forall x \in \mathbb{R}, $$	$$ tanh(x) $$	$$ 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \frac{x^6}{6!} + \ ... \ + \frac{x^{2n}}{(2n)!} + o(x^{2n+1})$$	$$ \sum_{k=0}^n \frac{x^{2k}}{(2k)!} + o(x^{2n+1}) $$
$$ \forall x \in \hspace{0.05em} ]-1, \hspace{0.1em} 1[, $$	$$ tanh(x) $$	$$ x - \frac{1}{3}x^3 + \frac{2}{15}x^4 - \frac{17}{315}x^6 + \ ... \ + o(x^{6})$$	$$ $$
$$\forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \Bigl\{ \frac{\pi}{2} + k\pi \Bigr\} \biggr], $$	$$ csc\left(\frac{\pi}{2} + x \right) = sec(x) $$	$$ 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + 61 \frac{x^6}{6!} + \ ... \ + o(x^{6})$$	$$ $$
$$\forall k \in \mathbb{Z}, \enspace \forall x \in \biggl[ \mathbb{R} \hspace{0.2em} \backslash \Bigl\{ \frac{\pi}{2} + k\pi \Bigr\} \biggr], $$	$$ cot\left(\frac{\pi}{2} + x \right) = -tan(x)$$	$$ -x -\frac{1}{3}x^3 -\frac{2}{15}x^4 -\frac{17}{315}x^6 + \ ... \ + o(x^{6})$$	$$ $$

Example of Taylor series

The sinus function$: f(x) = sin(x) $

Taylor series of order 3 at zero$: T_3(0) $

Let us use the previously demonstrated method to calculate a Taylor series of the $ sin(x) $ function.

We firstly verify that $ sin(x) $ can be derivated three times in a row. It is well-known that is the case.

Then, let us calculate the successive derivatives of order $3 $, and retrieve all of these images at $ a = 0 $.

$$f(x) = sin(x) \Longrightarrow sin(0) = 0 $$

$$f'(x) = cos(x) \Longrightarrow cos(0) = 1$$

$$f^{(2)}(x) = -sin(x) \Longrightarrow -sin(0) = 0$$

$$f^{(3)}(x) = -cos(x) \Longrightarrow -cos(0) = -1 $$

Now, we apply the Taylor-Young's formula.

$$ f_{n,a}(x) = f(a) + f'(a)(x - a) + \frac{f^{(2)}(a)}{2!}(x - a)^2 \hspace{0.2em} + \hspace{0.2em} ... \hspace{0.2em} + \hspace{0.2em} \frac{f^{(n)}(a)}{n!}(x - a)^n + R_{n, a}(x) $$

In our case, that would be:

$$ sin_{3,0}(x) = sin(0) + cos(0)(x - 0) - \frac{sin(0)}{2!}(x - 0)^2 - \frac{cos(0)}{3!}(x - 0)^3 + R_{3, 0}(x) $$

$$ sin_{3,0}(x) = x -\frac{1}{6}x^3 +R_{3, 0}(x) $$

Framing with integral remainder

$$ sin_{3,0}(x) = x -\frac{1}{6}x^3 +R_{3, 0}(x) $$

We have seen above that this remainder is worth:

$$ R_{3, 0}(x) = \int_0^x \frac{sin^{(4)}(t)}{3!}(x-t)^3 \hspace{0.2em} dt $$

But, $sin^{(4)}(t) = sin(t)$. So we now have a new expression for $ R_{3, 0}(x)$:

$$ R_{3, 0}(x) = \int_0^x \frac{sin(t)}{6}(x-t)^3 \hspace{0.2em} dt $$

Let us now frame this remainder in the interval $ [-\pi, \pi]$.

$$ -\pi \leqslant x \leqslant \pi$$

$$ -1 \leqslant sin(x) \leqslant 1$$

$$ \frac{-(x-t)^3}{6} \hspace{0.2em} \leqslant \hspace{0.2em} \frac{sin(x)}{6}(x-t)^3 \hspace{0.2em} \leqslant \hspace{0.2em} \frac{-(x-t)^3}{6} $$

Using the property of growth of an integral, we do have:

$$ -\int_a^x \frac{(x-t)^3}{6} \hspace{0.1em}dt \hspace{0.2em} \leqslant \hspace{0.2em} \int_a^x \frac{sin(x)}{6}(x-t)^3 \hspace{0.1em}dt \hspace{0.2em} \leqslant \hspace{0.2em} \int_a^x \frac{(x-t)^3}{6} \hspace{0.1em}dt$$

$$ -\Biggl[ \frac{(x-t)^4}{24} \Biggr]_0^x \hspace{0.2em} \leqslant \hspace{0.2em} R_{3, 0}(x) \hspace{0.2em} \leqslant \hspace{0.2em} \Biggl[ \frac{(x-t)^4}{24} \Biggr]_0^x $$

$$ -\frac{x^4}{24} \hspace{0.2em} \leqslant \hspace{0.2em} R_{3, 0}(x) \hspace{0.2em} \leqslant \hspace{0.2em} \frac{x^4}{24} $$

Yet, we know thanks to $(1)$ that:

$$ sin_{3, 0}(x) = \enspace T_{3,0}(x) + R_{3,0}(x) $$

This leads us to a framing for $ sin_{3, 0}(x)$ :

$$ T_{3, 0}(x) -\frac{x^4}{24} \hspace{0.2em} \leqslant \hspace{0.2em} T_{3, 0}(x) + R_{3, 0}(x) \hspace{0.2em} \leqslant \hspace{0.2em} T_{3, 0}(x) + \frac{x^4}{24} $$

$$ x -\frac{1}{6}x^3 -\frac{x^4}{24} \hspace{0.2em} \leqslant \hspace{0.2em} sin_{3, 0}(x) \hspace{0.2em} \leqslant \hspace{0.2em} x -\frac{1}{6}x^3 + \frac{x^4}{24} $$

Taylor series of order n at zero$: T_n(0) $

Performing an Taylor series of order $n$ at $(x=0)$ for the $sin(x)$ function, we obtain:

$$ sin_{n,0}(x) = x -\frac{x^3 }{3!}+ \frac{x^5}{5!} - \frac{x^7}{7!} \hspace{0.2em} + \hspace{0.2em} ... \hspace{0.2em} + \hspace{0.2em} (-1)^{n} \frac{x^{2n+1}}{(2n+1)!} + R_{n, 0}(x) $$

Moreover, the remainder of this Taylor series is worth:

$$ R_{n, 0}(x) = \int_0^x \frac{sin^{(n+1)}(t)}{(2n+1)!}(x-0)^{2n+1} \hspace{0.2em} dt $$

$$ R_{n, 0}(x) = \int_0^x \frac{cos(t)}{(2n+1)!}x^{2n+1} \hspace{0.2em} dt $$

With the Taylor-Lagrange's inequality, we can frame this remainder:

$$ \Bigl|R_{n, 0}(x) \Bigr| \leqslant M \Biggl | \frac{x^{2n+2}}{(2n+2)!} \Biggr|$$ $$ with \enspace M = max \Bigl\{cos(0) ; \hspace{0.2em}cos(x) \Bigr\} = 1 $$

$$ lim_{n \to +\infty} \enspace \Bigl |R_{n, 0}(x) \Bigr| \hspace{0.2em} \leqslant \hspace{0.2em} lim_{n \to +\infty} \enspace \Biggl | \frac{x^{2n+2}}{(2n+2)!} \Biggr| $$

By compared growth of the limits, the factorial function outweighs the power of x function:

$$ lim_{n \to +\infty} \enspace \Bigl |R_{n, 0}(x) \Bigr| \hspace{0.2em} \leqslant 0 $$

In the end, with the squeeze theorem:

$$ lim_{n \to +\infty} \enspace R_{n, 0}(x) = 0 $$

So as a result, a Taylor series of the $sin(x) $ function is worth:

$$ sin(x) = x -\frac{x^3 }{3!}+ \frac{x^5}{5!} - \frac{x^7}{7!} \hspace{0.2em} + \hspace{0.2em} ... \hspace{0.2em} + \hspace{0.2em} (-1)^{n} \frac{x^{2n+1}}{(2n+1)!} + R_{n, 0}(x) \qquad (with \enspace lim_{n \to +\infty} \enspace R_{n, 0}(x) = 0) $$

The Taylor-Young's formula with integral remainder

Demonstration

The Taylor-Young's formula

Recap of the main Taylor series

Example of Taylor series

The sinus function\(: f(x) = sin(x) \)

Taylor series of order 3 at zero\(: T_3(0) \)

Framing with integral remainder

Taylor series of order n at zero\(: T_n(0) \)