grandes-ecoles

Q1 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $f : [ a , b ] \rightarrow \mathbb { R }$ be a piecewise continuous function taking values in an interval $J$. Let $\varphi$ be a continuous and convex function on $J$. Prove that
$$\varphi \left( \frac { 1 } { b - a } \int _ { a } ^ { b } f ( t ) \mathrm { d } t \right) \leqslant \frac { 1 } { b - a } \int _ { a } ^ { b } \varphi \circ f ( t ) \mathrm { d } t .$$
You may use Riemann sums.

Q2 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View

Let $f : \mathbb { R } _ { + } \rightarrow \mathbb { R }$, be a piecewise continuous function, strictly positive and integrable. For all $x > 0$, we define
$$g ( x ) = \frac { 1 } { x } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t \quad \text { and } \quad h ( x ) = \frac { 1 } { x } g ( x ) = \frac { 1 } { x ^ { 2 } } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t .$$
Determine the limit of $g ( x )$ as $x$ tends to 0.

Q3 Indefinite & Definite Integrals Properties of Integral-Defined Functions (Continuity, Differentiability) View

Let $f : \mathbb { R } _ { + } \rightarrow \mathbb { R }$, be a piecewise continuous function, strictly positive and integrable. For all $x > 0$, we define
$$g ( x ) = \frac { 1 } { x } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t \quad \text { and } \quad h ( x ) = \frac { 1 } { x } g ( x ) = \frac { 1 } { x ^ { 2 } } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t .$$
Determine the limit of $g ( x )$ as $x$ tends to $+ \infty$. Denoting by $\mathbb { 1 } _ { [ 0 , x ] }$ the indicator function of $[ 0 , x ]$, you may note that $g ( x ) = \int _ { 0 } ^ { + \infty } \frac { 1 } { x } t f ( t ) \mathbb { 1 } _ { [ 0 , x ] } ( t ) \mathrm { d } t$.

Q4 Reduction Formulae Derive a Reduction/Recurrence Formula via Integration by Parts View

Let $f : \mathbb { R } _ { + } \rightarrow \mathbb { R }$, be a piecewise continuous function, strictly positive and integrable. For all $x > 0$, we define
$$g ( x ) = \frac { 1 } { x } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t \quad \text { and } \quad h ( x ) = \frac { 1 } { x } g ( x ) = \frac { 1 } { x ^ { 2 } } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t .$$
Deduce that the integral $\int _ { 0 } ^ { + \infty } h ( x ) \mathrm { d } x$ converges and that
$$\int _ { 0 } ^ { + \infty } f ( x ) \mathrm { d } x = \int _ { 0 } ^ { + \infty } h ( x ) \mathrm { d } x .$$
You may use integration by parts.

Q5 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $f$ be a piecewise continuous function, strictly positive, integrable on $\mathbb { R } _ { + }$. Prove that, for all $x > 0$,
$$\exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right) \leqslant \frac { \mathrm { e } } { x ^ { 2 } } \int _ { 0 } ^ { x } t f ( t ) \mathrm { d } t$$
You may note that $\ln ( f ( t ) ) = \ln ( t f ( t ) ) - \ln ( t )$.

Q6 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Let $f$ be a piecewise continuous function, strictly positive, integrable on $\mathbb { R } _ { + }$. Deduce that $x \mapsto \exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right)$ is integrable on $\mathbb { R } _ { + } ^ { * }$ and that
$$\int _ { 0 } ^ { + \infty } \exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right) \mathrm { d } x \leqslant \mathrm { e } \int _ { 0 } ^ { + \infty } f ( x ) \mathrm { d } x$$

Q8 Reduction Formulae Bound or Estimate a Parametric Integral View

We assume that $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ is a decreasing sequence of strictly positive real numbers. We denote by $f$ the step function which, for all $k \in \mathbb { N } ^ { * }$, equals $a _ { k }$ on the interval $[ k - 1 , k [$.
Prove that, for all $k$ in $\mathbb { N } ^ { * }$,
$$\int _ { k - 1 } ^ { k } \exp \left( \frac { 1 } { x } \int _ { 0 } ^ { x } \ln ( f ( t ) ) \mathrm { d } t \right) \mathrm { d } x \geqslant \exp \left( \frac { 1 } { k } \sum _ { i = 1 } ^ { k } \ln \left( a _ { i } \right) \right)$$
You may use the previous question.

Q9 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

We assume that $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ is a decreasing sequence of strictly positive real numbers. We denote by $f$ the step function which, for all $k \in \mathbb { N } ^ { * }$, equals $a _ { k }$ on the interval $[ k - 1 , k [$.
Deduce Carleman's inequality in the case where $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ is a decreasing sequence.

Q10 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Explain how one can remove the decreasing hypothesis in Carleman's inequality.

Q11 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$.
Let $s > 0$. We define the functions $f$ and $g _ { s }$ on $\overline { U _ { n } }$ by setting, for all $x = \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } }$,
$$f ( x ) = \prod _ { k = 1 } ^ { n } x _ { k } \quad \text { and } \quad g _ { s } ( x ) = \left( \sum _ { k = 1 } ^ { n } x _ { k } \right) - s .$$
We denote by $X _ { s }$ the subset of $\overline { U _ { n } }$ consisting of the zeros of $g _ { s } : X _ { s } = \left\{ x \in \overline { U _ { n } } \mid g _ { s } ( x ) = 0 \right\}$.
We admit that $f$ and $g _ { s }$ are of class $\mathcal { C } ^ { 1 }$ on $U _ { n }$. Give the expression of their gradient at a point $x = \left( x _ { 1 } , \ldots , x _ { n } \right)$ of $U _ { n }$.

Q12 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$.
Let $s > 0$. We define the functions $f$ and $g _ { s }$ on $\overline { U _ { n } }$ by setting, for all $x = \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } }$,
$$f ( x ) = \prod _ { k = 1 } ^ { n } x _ { k } \quad \text { and } \quad g _ { s } ( x ) = \left( \sum _ { k = 1 } ^ { n } x _ { k } \right) - s .$$
We denote by $X _ { s }$ the subset of $\overline { U _ { n } }$ consisting of the zeros of $g _ { s } : X _ { s } = \left\{ x \in \overline { U _ { n } } \mid g _ { s } ( x ) = 0 \right\}$.
Prove that the restriction of $f$ to $X _ { s }$ admits a maximum on $X _ { s }$ and that this maximum is in fact attained on $X _ { s } \cap U _ { n }$.
You may verify that $f$ is strictly positive at certain points of $X _ { s } \cap U _ { n }$.

Q13 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$.
Let $s > 0$. We define the functions $f$ and $g _ { s }$ on $\overline { U _ { n } }$ by setting, for all $x = \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } }$,
$$f ( x ) = \prod _ { k = 1 } ^ { n } x _ { k } \quad \text { and } \quad g _ { s } ( x ) = \left( \sum _ { k = 1 } ^ { n } x _ { k } \right) - s .$$
We denote by $X _ { s }$ the subset of $\overline { U _ { n } }$ consisting of the zeros of $g _ { s } : X _ { s } = \left\{ x \in \overline { U _ { n } } \mid g _ { s } ( x ) = 0 \right\}$.
We denote by $a = \left( a _ { 1 } , \ldots , a _ { n } \right)$ an element of $X _ { s } \cap U _ { n }$ at which the restriction of $f$ to $X _ { s }$ attains its maximum.
Prove that there exists a real number $\lambda > 0$ such that, for all $k$ in $\llbracket 1 , n \rrbracket , a _ { k } = \frac { f ( a ) } { \lambda }$.

Q14 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$.
Let $s > 0$. We define the functions $f$ and $g _ { s }$ on $\overline { U _ { n } }$ by setting, for all $x = \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } }$,
$$f ( x ) = \prod _ { k = 1 } ^ { n } x _ { k } \quad \text { and } \quad g _ { s } ( x ) = \left( \sum _ { k = 1 } ^ { n } x _ { k } \right) - s .$$
We denote by $X _ { s }$ the subset of $\overline { U _ { n } }$ consisting of the zeros of $g _ { s } : X _ { s } = \left\{ x \in \overline { U _ { n } } \mid g _ { s } ( x ) = 0 \right\}$.
Prove that, for all $\left( x _ { 1 } , \ldots , x _ { n } \right) \in U _ { n } \cap X _ { s } , \left( \prod _ { i = 1 } ^ { n } x _ { i } \right) ^ { 1 / n } \leqslant \frac { 1 } { n } \sum _ { i = 1 } ^ { n } x _ { i }$ and deduce the arithmetic-geometric inequality
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \left( \mathbb { R } _ { + } \right) ^ { n } , \quad \left( \prod _ { i = 1 } ^ { n } x _ { i } \right) ^ { 1 / n } \leqslant \frac { 1 } { n } \sum _ { i = 1 } ^ { n } x _ { i }$$

Q15 Applied differentiation Partial derivatives and multivariable differentiation View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $h _ { n }$ the map from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad h _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \cdots + x _ { n } - 1 .$$
We admit that $F _ { n }$ and $h _ { n }$ are both of class $\mathcal { C } ^ { 1 }$ on $U _ { n }$. We denote by $H _ { n }$ the set $H _ { n } = \left\{ \left( x _ { 1 } , \ldots , x _ { n } \right) \in \mathbb { R } ^ { n } \mid x _ { 1 } + \cdots + x _ { n } = 1 \right\}$.
Determine the gradient of $F _ { n }$ and the gradient of $h _ { n }$ at every point of $U _ { n }$.

Q16 Applied differentiation Partial derivatives and multivariable differentiation View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $H _ { n }$ the set $H _ { n } = \left\{ \left( x _ { 1 } , \ldots , x _ { n } \right) \in \mathbb { R } ^ { n } \mid x _ { 1 } + \cdots + x _ { n } = 1 \right\}$.
Prove that the restriction of $F _ { n }$ to $\overline { U _ { n } } \cap H _ { n }$ admits a maximum.

Q17 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $h _ { n }$ the map from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad h _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \cdots + x _ { n } - 1 .$$
We denote by $H _ { n }$ the set $H _ { n } = \left\{ \left( x _ { 1 } , \ldots , x _ { n } \right) \in \mathbb { R } ^ { n } \mid x _ { 1 } + \cdots + x _ { n } = 1 \right\}$. We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$.
Prove that there exists a real number $\lambda > 0$ such that
$$\left\{ \begin{aligned} \gamma _ { 1 } + \frac { \gamma _ { 2 } } { 2 } + \cdots + \frac { \gamma _ { n } } { n } & = \lambda a _ { 1 } \\ \frac { \gamma _ { 2 } } { 2 } + \cdots + \frac { \gamma _ { n } } { n } & = \lambda a _ { 2 } \\ & \vdots \\ \frac { \gamma _ { n } } { n } & = \lambda a _ { n } \\ a _ { 1 } + a _ { 2 } + \cdots + a _ { n } & = 1 \end{aligned} \right.$$

Q18 Stationary points and optimisation Existence or properties of extrema via abstract/theoretical argument View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. Its closure, denoted $\overline { U _ { n } }$, is $\left( \mathbb { R } _ { + } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$. There exists a real number $\lambda > 0$ satisfying the system in Q17.
Deduce that:
a) $\lambda = \gamma _ { 1 } + \gamma _ { 2 } + \cdots + \gamma _ { n } = M _ { n }$;
b) for all $k$ in $\llbracket 1 , n \rrbracket , \gamma _ { k } = \lambda \omega _ { k } a _ { k }$, where
$$\left\{ \begin{array} { l } \omega _ { k } = k \left( 1 - \frac { a _ { k + 1 } } { a _ { k } } \right) \text { if } k \in \llbracket 1 , n - 1 \rrbracket \\ \omega _ { n } = n \end{array} \right.$$

Q19 Proof Direct Proof of an Inequality View

The objective of this question is part of proving that $\lambda \leqslant \mathrm { e }$. We assume by contradiction that $\lambda > \mathrm { e }$.
Verify that, for all $k$ in $\mathbb { N } , \frac { 1 } { \mathrm { e } } \leqslant \left( \frac { k + 1 } { k + 2 } \right) ^ { k + 1 }$.

Q20 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$. We assume by contradiction that $\lambda > \mathrm { e }$, where $\lambda$ is the real number from Q17, and $\omega_k$ is as defined in Q18b.
Prove that $\omega _ { 1 } \leqslant \frac { 1 } { \mathrm { e } }$ and that, for all $k$ in $\llbracket 1 , n \rrbracket , \omega _ { k } \leqslant \frac { k } { k + 1 }$.
You may prove, for $k \in \llbracket 1 , n - 1 \rrbracket$, that $\omega _ { k + 1 } ^ { k + 1 } = \frac { 1 } { \lambda } \omega _ { k } ^ { k } \left( 1 - \frac { \omega _ { k } } { k } \right) ^ { - k }$.

Q21 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $n$ be in $\mathbb { N } ^ { * }$. We denote by $U _ { n }$ the open set $\left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$. We consider the map $F _ { n }$ from $\overline { U _ { n } }$ to $\mathbb { R }$, defined by
$$\forall \left( x _ { 1 } , \ldots , x _ { n } \right) \in \overline { U _ { n } } , \quad F _ { n } \left( x _ { 1 } , \ldots , x _ { n } \right) = x _ { 1 } + \left( x _ { 1 } x _ { 2 } \right) ^ { 1 / 2 } + \left( x _ { 1 } x _ { 2 } x _ { 3 } \right) ^ { 1 / 3 } + \cdots + \left( x _ { 1 } \cdots x _ { n } \right) ^ { 1 / n } .$$
We denote by $M _ { n }$ the maximum of $F _ { n }$ on $\overline { U _ { n } } \cap H _ { n }$ and we denote by $( a _ { 1 } , \ldots , a _ { n } )$ a point of $U _ { n } \cap H _ { n }$ at which it is attained. For $k$ between 1 and $n$, we denote $\gamma _ { k } = \left( a _ { 1 } a _ { 2 } \cdots a _ { k } \right) ^ { 1 / k }$. We assume by contradiction that $\lambda > \mathrm { e }$, where $\lambda$ is the real number from Q17, and $\omega_k$ is as defined in Q18b.
Reach a contradiction on $\omega _ { n }$. Deduce that, for all $n$ in $\mathbb { N } ^ { * }$, for all $\left( x _ { 1 } , \ldots , x _ { n } \right) \in \left( \mathbb { R } _ { + } ^ { * } \right) ^ { n }$ such that $x _ { 1 } + \cdots + x _ { n } = 1$,
$$\sum _ { k = 1 } ^ { n } \left( x _ { 1 } x _ { 2 } \cdots x _ { k } \right) ^ { 1 / k } \leqslant \mathrm { e }$$

Q22 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Deduce Carleman's inequality:
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \mathrm { e } \sum _ { n = 1 } ^ { + \infty } a _ { n }$$
for any sequence $\left( a _ { k } \right) _ { k \in \mathbb { N } ^ { * } }$ of strictly positive real numbers such that $\sum a _ { n }$ converges.

Q23 Taylor series Limit evaluation using series expansion or exponential asymptotics View

Let $\varphi$ be the function defined by
$$\forall t \in ] - 1,1 \left[ \backslash \{ 0 \} , \quad \varphi ( t ) = ( 1 - t ) ^ { 1 - 1 / t } \right.$$
Justify that $\varphi$ is extendable by continuity at 0 and specify the value of its extension at 0. We will still denote this extension by $\varphi$.

Q24 Sequences and series, recurrence and convergence Series convergence and power series analysis View

We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
Prove that, for all $n$ in $\mathbb { N } ^ { * } , \left| b _ { n } \right| \leqslant 1$. Deduce an inequality on the radius of convergence of the power series $\sum _ { k \geqslant 0 } b _ { k } t ^ { k }$.

Q25 Taylor series Recursive or implicit derivative computation for series coefficients View

Let $\varphi$ be the function defined by
$$\forall t \in ] - 1,1 \left[ \backslash \{ 0 \} , \quad \varphi ( t ) = ( 1 - t ) ^ { 1 - 1 / t } \right.$$
We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
Prove that, for all $t$ in $] - 1,1 \left[ , \varphi ^ { \prime } ( t ) = \varphi ( t ) \psi ( t ) \right.$, where
$$\forall t \in ] - 1,1 \left[ , \quad \psi ( t ) = - \sum _ { n = 0 } ^ { + \infty } \frac { 1 } { n + 2 } t ^ { n } \right.$$
then that, for all $n$ in $\mathbb { N } ^ { * }$,
$$\varphi ^ { ( n ) } ( 0 ) = - \sum _ { k = 0 } ^ { n - 1 } \frac { k ! } { k + 2 } \binom { n - 1 } { k } \varphi ^ { ( n - k - 1 ) } ( 0 )$$

Q26 Taylor series Formal power series manipulation (Cauchy product, algebraic identities) View

Let $\varphi$ be the function defined by
$$\forall t \in ] - 1,1 \left[ \backslash \{ 0 \} , \quad \varphi ( t ) = ( 1 - t ) ^ { 1 - 1 / t } \right.$$
We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
Conclude that
$$\forall t \in ] - 1,1 \left[ , \quad \varphi ( t ) = \mathrm { e } \left( 1 - \sum _ { k = 1 } ^ { + \infty } b _ { k } t ^ { k } \right) \right.$$

Q27 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ and $\left( c _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be two sequences of strictly positive real numbers. Prove that
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \sum _ { k = 1 } ^ { + \infty } c _ { k } a _ { k } \sum _ { n = k } ^ { + \infty } \frac { 1 } { n } \left( \prod _ { i = 1 } ^ { n } c _ { i } \right) ^ { - 1 / n }$$

Q28 Sequences and Series Recurrence Relations and Sequence Properties View

Let $\left( a _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ and $\left( c _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ be two sequences of strictly positive real numbers. We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
By considering $c _ { n } = \frac { ( n + 1 ) ^ { n } } { n ^ { n - 1 } }$, deduce the Carleman-Yang inequality:
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \mathrm { e } \sum _ { n = 1 } ^ { + \infty } \left( 1 - \sum _ { k = 1 } ^ { + \infty } \frac { b _ { k } } { ( n + 1 ) ^ { k } } \right) a _ { n }$$

Q29 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

We define the sequence $\left( b _ { n } \right) _ { n \in \mathbb { N } }$ by
$$\left\{ \begin{array} { l } b _ { 0 } = - 1 \\ \forall n \in \mathbb { N } ^ { * } , \quad b _ { n } = - \frac { 1 } { n } \sum _ { k = 1 } ^ { n } \frac { 1 } { k + 1 } b _ { n - k } \end{array} \right.$$
The Carleman-Yang inequality states:
$$\sum _ { n = 1 } ^ { + \infty } \left( \prod _ { k = 1 } ^ { n } a _ { k } \right) ^ { 1 / n } \leqslant \mathrm { e } \sum _ { n = 1 } ^ { + \infty } \left( 1 - \sum _ { k = 1 } ^ { + \infty } \frac { b _ { k } } { ( n + 1 ) ^ { k } } \right) a _ { n }$$
Prove that, for all $n$ in $\mathbb { N } ^ { * } , b _ { n } \geqslant 0$. In what way is the previous inequality a refinement of Carleman's inequality?

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