grandes-ecoles

QI.A.1 Factor & Remainder Theorem Polynomial Degree and Structural Properties View

For a nonzero polynomial $P \in \mathbb{R}_n[X]$, express $\operatorname{deg}(\tau(P))$ and $\operatorname{cd}(\tau(P))$ in terms of $\operatorname{deg}(P)$ and $\operatorname{cd}(P)$.

QI.A.2 Factor & Remainder Theorem Remainder Theorem with Composed or Shifted Arguments View

Let $P \in \mathbb{R}_n[X]$. For $k \in \mathbb{N}$, give the expression of $\tau^k(P)$ as a function of $P$.

QI.A.3 Matrices Matrix Entry and Coefficient Identities View

Give the matrix $M = \left(M_{i,j}\right)_{1 \leqslant i,j \leqslant n+1}$ of $\tau$ in the basis $\left(P_k\right)_{k \in \llbracket 1, n+1 \rrbracket}$. Express the coefficients $M_{i,j}$ in terms of $i$ and $j$.

QI.A.4 Invariant lines and eigenvalues and vectors Compute eigenvalues of a given matrix View

Specify the set of eigenvalues of $\tau$. Is the application $\tau$ diagonalizable?

QI.A.5 Matrices Linear Transformation and Endomorphism Properties View

Is the application $\tau$ bijective? If so, specify $\tau^{-1}$. Is the expression of $\tau^j$ found in question I.A.2 for $j \in \mathbb{N}$ valid for $j \in \mathbb{Z}$?

QI.A.6 Matrices Linear System and Inverse Existence View

What is $M^{-1}$? Express the coefficients $\left(M^{-1}\right)_{i,j}$ in terms of $i$ and $j$.

QI.A.7 Matrices Matrix Entry and Coefficient Identities View

We are given a real sequence $\left(u_k\right)_{k \in \mathbb{N}}$ and we define for every integer $k \in \mathbb{N}$ $$v_k = \sum_{j=0}^{k} \binom{k}{j} u_j$$ Determine a matrix $Q \in \mathcal{M}_{n+1}(\mathbb{R})$ such that $$\left(\begin{array}{c} v_0 \\ v_1 \\ \vdots \\ v_n \end{array}\right) = Q \left(\begin{array}{c} u_0 \\ u_1 \\ \vdots \\ u_n \end{array}\right)$$

QI.A.8 Proof Deduction or Consequence from Prior Results View

Deduce the inversion formula: for every integer $k \in \mathbb{N}$, $$u_k = \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} v_j$$

QI.A.9 Sequences and Series Functional Equations and Identities via Series View

We consider a real $\lambda$ and the sequence $\left(u_k = \lambda^k\right)_{k \in \mathbb{N}}$. What is the sequence $\left(v_k\right)_{k \in \mathbb{N}}$ defined by formula $$v_k = \sum_{j=0}^{k} \binom{k}{j} u_j \quad \text{(I.1)}$$ Then verify formula $$u_k = \sum_{j=0}^{k} (-1)^{k-j} \binom{k}{j} v_j \quad \text{(I.2)}$$

QI.B.1 Factor & Remainder Theorem Polynomial Degree and Structural Properties View

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ For a non-constant polynomial $P \in \mathbb{R}_n[X]$, express $\operatorname{deg}(\delta(P))$ and $\operatorname{cd}(\delta(P))$ in terms of $\operatorname{deg}(P)$ and $\operatorname{cd}(P)$.

QI.B.2 Factor & Remainder Theorem Proof of Polynomial Divisibility or Identity View

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ Deduce the kernel $\operatorname{ker}(\delta)$ and the image $\operatorname{Im}(\delta)$ of the endomorphism $\delta$.

QI.B.3 Factor & Remainder Theorem Proof of Polynomial Divisibility or Identity View

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ More generally, for $j \in \llbracket 1, n \rrbracket$, show the following equalities: $$\operatorname{ker}\left(\delta^j\right) = \mathbb{R}_{j-1}[X] \quad \text{and} \quad \operatorname{Im}\left(\delta^j\right) = \mathbb{R}_{n-j}[X]$$

QI.B.4 Factor & Remainder Theorem Proof of Polynomial Divisibility or Identity View

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ For $k \in \mathbb{N}$ and $P \in \mathbb{R}_n[X]$, express $\delta^k(P)$ in terms of $\tau^j(P)$ for $j \in \llbracket 0, k \rrbracket$.

QI.B.5 Proof Direct Proof of a Stated Identity or Equality View

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ Let $P \in \mathbb{R}_{n-1}[X]$. Show that $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} P(j) = 0$$

QI.B.6 Matrices Matrix Algebra and Product Properties View

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ In this question, we propose to show that there does not exist a linear application $u : \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X]$ such that $u \circ u = \delta$. We suppose, by contradiction, that such an application $u$ exists.
a) Show that $u$ and $\delta^2$ commute.
b) Deduce that $\mathbb{R}_1[X]$ is stable under the application $u$.
c) Show that there does not exist a matrix $A \in \mathcal{M}_2(\mathbb{R})$ such that $$A^2 = \left(\begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right)$$
d) Conclude.

QI.B.7 Factor & Remainder Theorem Proof of Polynomial Divisibility or Identity View

The difference operator is the endomorphism $\delta$ of $\mathbb{R}_n[X]$ such that $\delta = \tau - \operatorname{Id}_{\mathbb{R}_n[X]}$: $$\delta : \left\{ \begin{array}{l} \mathbb{R}_n[X] \rightarrow \mathbb{R}_n[X] \\ P(X) \mapsto P(X+1) - P(X) \end{array} \right.$$ In this question, we seek all vector subspaces of $\mathbb{R}_n[X]$ stable under the application $\delta$.
a) For a nonzero polynomial $P$ of degree $d \leqslant n$, show that the family $(P, \delta(P), \ldots, \delta^d(P))$ is free. What is the vector space spanned by this family?
b) Deduce that if $V$ is a vector subspace of $\mathbb{R}_n[X]$ stable under $\delta$ and not reduced to $\{0\}$, there exists an integer $d \in \llbracket 0, n \rrbracket$ such that $V = \mathbb{R}_d[X]$.

QII.A.1 Permutations & Arrangements Counting Functions with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
What is $S(p, n)$ for $p < n$?

QII.A.2 Permutations & Arrangements Counting Functions with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
Determine $S(n, n)$.

QII.A.3 Combinations & Selection Counting Functions or Mappings with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
Determine $S(n+1, n)$.

QII.B.1 Combinations & Selection Counting Functions or Mappings with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
How many applications are there from $\llbracket 1, p \rrbracket$ to $\llbracket 1, n \rrbracket$?

QII.B.2 Combinations & Selection Counting Functions or Mappings with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
For $p \geqslant n$, establish the formula $$n^p = \sum_{k=0}^{n} \binom{n}{k} S(p, k)$$ where $S(p, 0) = 0$ by convention.

QII.B.3 Combinations & Selection Counting Functions or Mappings with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
Deduce an expression of $S(p, n)$ for $p \geqslant n$.

QII.B.4 Combinations & Selection Counting Functions or Mappings with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
By rereading question I.B.5, comment on the consistency of the expression of $S(p,n)$ for $p < n$.

QII.C Combinations & Selection Counting Functions or Mappings with Constraints View

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
Simplify as much as possible the following expressions: $$\sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} k^n \quad \text{and} \quad \sum_{k=0}^{n} (-1)^{n-k} \binom{n}{k} k^{n+1}$$

QIII.A.1 Proof Proof of Set Membership, Containment, or Structural Property View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that the family $\left(H_k\right)_{k \in \llbracket 0, n \rrbracket}$ is a basis of $\mathbb{R}_n[X]$.

QIII.A.2 Proof Computation of a Limit, Value, or Explicit Formula View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Calculate $\delta\left(H_0\right)$ and, for $k \in \llbracket 1, n \rrbracket$, express $\delta\left(H_k\right)$ in terms of $H_{k-1}$.

QIII.A.3 Invariant lines and eigenvalues and vectors Diagonalizability determination or proof View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Is the matrix $M$ defined in question I.A.3 and the matrix $M'$ of size $n+1$ given by $$M' = \left(\begin{array}{ccccc} 1 & 1 & 0 & \ldots & 0 \\ 0 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 1 \\ 0 & \ldots & \ldots & 0 & 1 \end{array}\right)$$ similar?

QIII.A.4 Proof Direct Proof of a Stated Identity or Equality View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that, for $k, l \in \llbracket 0, n \rrbracket$, $$\delta^k\left(H_l\right)(0) = \begin{cases} 1 & \text{if } k = l \\ 0 & \text{if } k \neq l \end{cases}$$

QIII.A.5 Proof Direct Proof of a Stated Identity or Equality View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that, for every $P \in \mathbb{R}_n[X]$, $$P = \sum_{k=0}^{n} \left(\delta^k(P)\right)(0) H_k$$

QIII.B.1 Proof Computation of a Limit, Value, or Explicit Formula View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Give the coordinates of the polynomial $X^3 + 2X^2 + 5X + 7$ in the basis $(H_0, H_1, H_2, H_3)$ of $\mathbb{R}_3[X]$.

QIII.B.2 Proof Deduction or Consequence from Prior Results View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Deduce a polynomial $P \in \mathbb{R}_5[X]$ such that $$\delta^2(P) = X^3 + 2X^2 + 5X + 7$$

QIII.B.3 Proof Characterization or Determination of a Set or Class View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Determine the real sequences $\left(u_k\right)_{k \in \mathbb{N}}$ such that $$u_{k+2} - 2u_{k+1} + u_k = k^3 + 2k^2 + 5k + 7 \quad (k \in \mathbb{N})$$

QIII.C.1 Proof Computation of a Limit, Value, or Explicit Formula View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Let $k \in \mathbb{Z}$. Calculate $H_n(k)$. Distinguish three cases: $k \in \llbracket 0, n-1 \rrbracket$, $k \geqslant n$, and $k < 0$. For the latter case, set $k = -p$.

QIII.C.2 Proof Deduction or Consequence from Prior Results View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Deduce that $H_n(\mathbb{Z}) \subset \mathbb{Z}$, that is, $H_n$ is integer-valued on the integers.

QIII.C.3 Proof Proof That a Map Has a Specific Property View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Let $P \in \mathbb{R}_n[X]$ be integer-valued on the integers. Show that $\delta(P)$ is also integer-valued on the integers.

QIII.C.4 Proof Proof of Equivalence or Logical Relationship Between Conditions View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Show that $P \in \mathbb{R}_n[X]$ is integer-valued on the integers if and only if its coordinates in the basis $\left(H_k\right)_{k \in \llbracket 0, n \rrbracket}$ are integers.

QIII.C.5 Proof Deduction or Consequence from Prior Results View

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Let $P \in \mathbb{R}[X]$ of degree $d \in \mathbb{N}$. Show that if $P$ is integer-valued on the integers then $d! P$ is a polynomial with integer coefficients. Study the converse.

QIV.A.1 Chain Rule Proof of Differentiability Class for Parameterized Integrals View

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Show that $\delta(f)$ is of class $\mathcal{C}^\infty$ on $\mathbb{R}_+^*$. Compare $(\delta(f))'$ and $\delta(f')$.

QIV.A.2 Binomial Theorem (positive integer n) Evaluate a Summation Involving Binomial Coefficients View

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
For $n \in \mathbb{N}$ and $x > 0$, express $\left(\delta^n(f)\right)(x)$ using the binomial coefficients $\binom{n}{j}$ and the $f(x+j)$ (where the index $j$ belongs to $\llbracket 0, n \rrbracket$).

QIV.A.3 Chain Rule Limit Evaluation Involving Composition or Substitution View

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Explain why, for every $x > 0$, there exists a $y_1 \in \left]0, 1\right[$ such that $$(\delta(f))(x) = f'(x + y_1)$$

QIV.A.4 Chain Rule Proof of Differentiability Class for Parameterized Integrals View

For an application $f : \mathbb{R}_+^* \rightarrow \mathbb{R}$ of class $\mathcal{C}^\infty$, we define the application $$\delta(f) : \left\{ \begin{array}{l} \mathbb{R}_+^* \rightarrow \mathbb{R} \\ x \mapsto f(x+1) - f(x) \end{array} \right.$$
Deduce that for every $x > 0$, for every $n \in \mathbb{N}^*$, there exists a $y_n \in \left]0, n\right[$ such that $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ One may proceed by induction on $n \in \mathbb{N}^*$ and use the three preceding questions.

QIV.B.1 Proof Deduction or Consequence from Prior Results View

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
Show that for every integer $k$ strictly positive, $k^\alpha$ belongs to $\mathbb{N}^*$.

QIV.B.2 Proof Direct Proof of an Inequality View

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
Show that $\alpha$ is positive or zero.

QIV.B.3 Differentiating Transcendental Functions Higher-order or nth derivative computation View

We consider throughout the rest of this part a real $\alpha$. We assume that for every prime number $p$, $p^\alpha$ is a natural number. We propose to show that $\alpha$ is then a natural number.
We consider the application $f_\alpha$ defined on $\mathbb{R}_+^*$ by $f_\alpha(x) = x^\alpha$. Show that $\alpha$ is a natural number if and only if one of the successive derivatives of $f_\alpha$ vanishes at least at one strictly positive real.

QIV.C.1 Proof Proof Involving Combinatorial or Number-Theoretic Structure View

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$ (where $\lfloor \cdot \rfloor$ denotes the floor function). We now choose $x \in \mathbb{N}^*$.
Show that the expression $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f_\alpha(x+j)$$ is a relative integer.

QIV.C.2 Differentiating Transcendental Functions Limit involving transcendental functions View

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$. The notations are those of question IV.A.4.
What is the limit of the expression $f_\alpha^{(n)}(x + y_n)$ when $x \in \mathbb{N}^*$ tends to $+\infty$?

QIV.C.3 Proof Deduction or Consequence from Prior Results View

We consider a real $\alpha$ such that for every prime number $p$, $p^\alpha$ is a natural number. We apply relation $$\sum_{j=0}^{n} (-1)^{n-j} \binom{n}{j} f(x+j) = f^{(n)}(x + y_n) \quad \text{(IV.1)}$$ to the function $f_\alpha(x) = x^\alpha$ and to the integer $n = \lfloor \alpha \rfloor + 1$.
Conclude that $\alpha$ is a natural number.

grandes-ecoles

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