grandes-ecoles

QI.A.1 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View

Show that $\sum _ { k = 0 } ^ { n } \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } = 1$.

QI.A.2 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View

Show that $\sum _ { k = 0 } ^ { n } k \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } = n x$.

QI.A.3 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View

Show that $\sum _ { k = 0 } ^ { n } k ( k - 1 ) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } = n ( n - 1 ) x ^ { 2 }$.

QI.A.4 Binomial Theorem (positive integer n) Prove a Binomial Identity or Inequality View

Deduce from the previous questions that $$\sum _ { k = 0 } ^ { n } \left( x - \frac { k } { n } \right) ^ { 2 } \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } = \frac { x ( 1 - x ) } { n } .$$

QI.B.1 Binomial Theorem (positive integer n) Multi-Part Structured Problem Involving Binomial Expansions View

Let $n \in \mathbb { N } ^ { * }$ and $x \in [ 0,1 ]$. We consider the sum $$S ( x ) = \sum _ { k = 0 } ^ { n } \left| x - \frac { k } { n } \right| \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } .$$ We denote

$V$ the set of integers $k \in \{ 0 , \ldots , n \}$ such that $\left| x - \frac { k } { n } \right| \leqslant \frac { 1 } { \sqrt { n } }$,
$W$ the set of integers $k \in \{ 0 , \ldots , n \}$ such that $\left| x - \frac { k } { n } \right| > \frac { 1 } { \sqrt { n } }$, and we set

$$S _ { V } ( x ) = \sum _ { k \in V } \left| x - \frac { k } { n } \right| \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } \quad \text { and } \quad S _ { W } ( x ) = \sum _ { k \in W } \left| x - \frac { k } { n } \right| \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } .$$ a) Show that $S _ { V } ( x ) \leqslant \frac { 1 } { \sqrt { n } }$. b) Show that $S _ { W } ( x ) \leqslant \frac { x ( 1 - x ) } { \sqrt { n } }$. c) Deduce that $S ( x ) \leqslant \frac { 5 } { 4 \sqrt { n } }$.

QI.B.2 Binomial Theorem (positive integer n) Multi-Part Structured Problem Involving Binomial Expansions View

Let $n \in \mathbb { N } ^ { * }$ and $x \in [ 0,1 ]$. We consider the sum $$S ( x ) = \sum _ { k = 0 } ^ { n } \left| x - \frac { k } { n } \right| \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k } .$$ a) Write the Cauchy-Schwarz inequality in the space $\mathbb { R } ^ { n + 1 }$ equipped with its canonical inner product. b) Using question I.A.4, deduce that $S ( x ) \leqslant \frac { 1 } { 2 \sqrt { n } }$.

QI.C.1 Binomial Theorem (positive integer n) Multi-Part Structured Problem Involving Binomial Expansions View

We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by $$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$ for all $x \in [ 0,1 ]$.
If $f ( x ) = x ^ { 2 }$ for all $x \in [ 0,1 ]$, determine, for all $n \in \mathbb { N } ^ { * }$, the polynomial $B _ { n } ( f )$ and deduce the value of $\left\| B _ { n } ( f ) - f \right\| _ { \infty }$.

QI.C.2 Binomial Theorem (positive integer n) Multi-Part Structured Problem Involving Binomial Expansions View

We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by $$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$ for all $x \in [ 0,1 ]$.
Let $f \in \mathcal { C }$. Show, for all $x \in [ 0,1 ]$, the relation $$B _ { n } ( f ) ( x ) - f ( x ) = \sum _ { k = 0 } ^ { n } \left( f \left( \frac { k } { n } \right) - f ( x ) \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$

QI.C.3 Binomial Theorem (positive integer n) Multi-Part Structured Problem Involving Binomial Expansions View

We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by $$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$ for all $x \in [ 0,1 ]$.
a) Show that if $f$ is $\delta$-Lipschitz, then $\left\| B _ { n } ( f ) - f \right\| _ { \infty } \leqslant \frac { \delta } { 2 \sqrt { n } }$ for all integer $n \geqslant 1$. b) Deduce that if $f$ is of class $C ^ { 1 }$, then there exists a real $c$ such that, for all $n \in \mathbb { N } ^ { * } , \left\| B _ { n } ( f ) - f \right\| _ { \infty } \leqslant \frac { c } { \sqrt { n } }$. c) Extend the previous result to the case where $f$ is a continuous function, piecewise $C ^ { 1 }$.

QI.C.4 Binomial Theorem (positive integer n) Multi-Part Structured Problem Involving Binomial Expansions View

We denote by $\mathcal { C }$ the vector space of continuous functions from $[ 0,1 ]$ to $\mathbb { R }$, equipped with the supremum norm $\| f \| _ { \infty } = \sup _ { x \in [ 0,1 ] } | f ( x ) |$. For $f \in \mathcal { C }$ and $n \in \mathbb { N } ^ { * }$, the $n$-th Bernstein polynomial of $f$ is defined by $$B _ { n } ( f ) ( x ) = \sum _ { k = 0 } ^ { n } f \left( \frac { k } { n } \right) \binom { n } { k } x ^ { k } ( 1 - x ) ^ { n - k }$$ for all $x \in [ 0,1 ]$.
Let $f : [ 0,1 ] \rightarrow \mathbb { R }$ be a continuous function, piecewise $C ^ { 1 }$. Deduce from the above that, for all real $r > 0$, there exists a polynomial $P$ with real coefficients such that $\forall x \in [ 0,1 ] , f ( x ) - r \leqslant P ( x ) \leqslant f ( x ) + r$.

QII.A.1 Sequences and Series Power Series Expansion and Radius of Convergence View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence. We assume that the associated power series $\sum a _ { n } x ^ { n }$ has radius of convergence $R _ { a } = 1$ and that the sum $f$ of this series satisfies $f ( x ) \sim \frac { 1 } { 1 - x }$ when $x \rightarrow 1, x < 1$.
Determine a real sequence $\left( b _ { n } \right) _ { n \geqslant 0 }$ such that $$\forall x \in ] - 1,1 [ , \quad \frac { 1 } { 1 - x ^ { 2 } } = \sum _ { n = 0 } ^ { + \infty } b _ { n } x ^ { n }$$

QII.A.2 Sequences and Series Power Series Expansion and Radius of Convergence View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence. We assume that the associated power series $\sum a _ { n } x ^ { n }$ has radius of convergence $R _ { a } = 1$ and that the sum $f$ of this series satisfies $f ( x ) \sim \frac { 1 } { 1 - x }$ when $x \rightarrow 1, x < 1$.
Deduce an example of a sequence $\left( a _ { n } \right) _ { n \geqslant 0 }$ satisfying hypothesis II.1 ($f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$) but not converging to 1.

QII.B.1 Sequences and Series Power Series Expansion and Radius of Convergence View

Give the power series expansion of the function $t \mapsto \frac { 1 } { ( 1 - t ) ^ { 2 } }$ as well as its radius of convergence. Specify whether the series converges at the endpoints of the interval of convergence.

QII.B.2 Partial Fractions View

We consider the functions $\varphi : x \mapsto \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { 2 } }$ and $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$. Determine sequences $\left( u _ { n } \right) _ { n \in \mathbb { N } }$ and $\left( v _ { n } \right) _ { n \in \mathbb { N } }$ such that, for all $x \in ] - 1,1 [$, $$\varphi ( x ) = \sum _ { n = 0 } ^ { + \infty } u _ { n } x ^ { n } \quad \text { and } \quad \psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n } .$$ We will express explicitly as a function of $n$, according to the parity of $n$, the reals $u _ { n }$ and $v _ { n }$.

QII.B.3 Sequences and Series Evaluation of a Finite or Infinite Sum View

We consider $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$ with power series expansion $\psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n }$ for $x \in ] - 1,1 [$. We denote $A_n = \sum_{k=0}^n a_k$ and $\widetilde{a}_n = \frac{A_n}{n+1}$.
Calculate $\widetilde { v } _ { n }$ (arithmetic mean of the numbers $v _ { 0 } , \ldots , v _ { n }$).

QII.B.4 Sequences and Series Functional Equations and Identities via Series View

We consider $\psi : x \mapsto \frac { 1 } { ( 1 + x ) ^ { 2 } ( 1 - x ) }$ with power series expansion $\psi ( x ) = \sum _ { n = 0 } ^ { + \infty } v _ { n } x ^ { n }$ for $x \in ] - 1,1 [$. We denote $\widetilde{a}_n = \frac{1}{n+1}\sum_{k=0}^n a_k$.
Construct using $\psi$ an example of a sequence $\left( a _ { n } \right) _ { n \geqslant 0 }$ satisfying hypothesis II.1 ($f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$) but not satisfying property II.3 ($\lim_{n\to\infty} \widetilde{a}_n = 1$).

QII.C.1 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A _ { n } = \sum _ { k = 0 } ^ { n } a _ { k }$.
For all $x \in \left[ 0,1 \left[ \right. \right.$ and all $n \in \mathbb { N }$, show that $f ( x ) \geqslant A _ { n } x ^ { n }$.

QII.C.2 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$.
Show the existence of an integer $N > 0$ such that $$\forall n \geqslant N , \quad f \left( \mathrm { e } ^ { - 1 / n } \right) \leqslant \frac { 2 } { 1 - \mathrm { e } ^ { - 1 / n } }$$

QII.C.3 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A _ { n } = \sum _ { k = 0 } ^ { n } a _ { k }$ and $\widetilde { a } _ { n } = \frac { A _ { n } } { n + 1 }$.
Deduce that the sequence $\left( \widetilde { a } _ { n } \right) _ { n \geqslant 0 }$ is bounded above.

QII.D.1 Sequences and Series Properties and Manipulation of Power Series or Formal Series View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A _ { n } = \sum _ { k = 0 } ^ { n } a _ { k }$, $\widetilde { a } _ { n } = \frac { A _ { n } } { n + 1 }$, and $\mu > 0$ is an upper bound of the sequence $\left( \widetilde { a } _ { n } \right) _ { n \geqslant 0 }$: $\forall n \in \mathbb { N } , \widetilde { a } _ { n } \leqslant \mu$.
a) For all $x \in ] - 1,1 [$, show that $( 1 - x ) \sum _ { k = 0 } ^ { + \infty } A _ { k } x ^ { k } = f ( x )$. b) Deduce that for all $x \in \left[ 0,1 \left[ \right. \right.$ and all $N \in \mathbb { N } ^ { * }$ $$\frac { f ( x ) } { 1 - x } \leqslant A _ { N - 1 } \frac { 1 - x ^ { N } } { 1 - x } + \mu \sum _ { k = N } ^ { + \infty } ( k + 1 ) x ^ { k }$$ c) Deduce that for all $x \in \left[ 0,1 \left[ \right. \right.$ and all $N \in \mathbb { N } ^ { * }$ $$f ( x ) \leqslant A _ { N - 1 } + \mu \left( ( N + 1 ) x ^ { N } + \frac { x ^ { N + 1 } } { 1 - x } \right) .$$

QII.D.2 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A _ { n } = \sum _ { k = 0 } ^ { n } a _ { k }$, $\widetilde { a } _ { n } = \frac { A _ { n } } { n + 1 }$, and $\mu > 0$ is an upper bound of $\left( \widetilde { a } _ { n } \right)$.
Let $\lambda$ be a strictly positive real number. a) Show that there exists an integer $N _ { 0 } > 0$ such that for all $N \geqslant N _ { 0 }$, $$f \left( \mathrm { e } ^ { - \lambda / N } \right) \geqslant \frac { 1 } { 2 \left( 1 - \mathrm { e } ^ { - \lambda / N } \right) } \geqslant \frac { N } { 2 \lambda } .$$ b) Show that for all $N \geqslant N _ { 0 }$ $$\tilde { a } _ { N - 1 } \geqslant \frac { 1 } { 2 \lambda } - \mu \mathrm { e } ^ { - \lambda } \left( 1 + \frac { 1 } { N } + \mathrm { e } ^ { - \lambda / N } \frac { 1 } { N \left( 1 - \mathrm { e } ^ { - \lambda / N } \right) } \right)$$ c) Determine as a function of $\lambda$ the limit, when $N$ tends to infinity, of the right-hand side in the previous inequality. d) Show that there exists a real $\lambda > 0$ such that this limit is strictly positive.

QII.D.3 Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $\widetilde { a } _ { n } = \frac { A _ { n } } { n + 1 }$ where $A_n = \sum_{k=0}^n a_k$.
Conclude that there exists a real $\nu > 0$ such that from a certain rank onwards we have $\widetilde { a } _ { n } \geqslant \nu$.

QII.E.1 Indefinite & Definite Integrals Piecewise/Periodic Function Integration View

Let $g : [ 0,1 ] \rightarrow \mathbb { R }$ be the function such that $g ( x ) = 1 / x$ if $x \geqslant \mathrm { e } ^ { - 1 }$ and $g ( x ) = 0$ otherwise. We fix a real $\varepsilon \in ] 0 , \mathrm { e } ^ { - 1 } [$. We define two continuous applications $g ^ { + } , g ^ { - } : [ 0,1 ] \rightarrow \mathbb { R }$ as follows:

$g ^ { + }$ is affine on $\left[ \mathrm { e } ^ { - 1 } - \varepsilon , \mathrm { e } ^ { - 1 } \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } - \varepsilon \right] \cup \left[ \mathrm { e } ^ { - 1 } , 1 \right]$;
$g ^ { - }$ is affine on $\left[ \mathrm { e } ^ { - 1 } , \mathrm { e } ^ { - 1 } + \varepsilon \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } \left[ \cup \left[ \mathrm { e } ^ { - 1 } + \varepsilon , 1 \right] \right. \right.$.

Calculate $\int _ { 0 } ^ { 1 } g ^ { + } ( t ) \mathrm { d } t$ and $\int _ { 0 } ^ { 1 } g ^ { - } ( t ) \mathrm { d } t$.

QII.E.2 Sequences and Series Properties and Manipulation of Power Series or Formal Series View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$.
Let $P$ be a polynomial with real coefficients. Show that $$( 1 - x ) \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n } P \left( x ^ { n } \right) \underset { \substack { x \rightarrow 1 \\ x < 1 } } { \longrightarrow } \int _ { 0 } ^ { 1 } P ( t ) \mathrm { d } t$$ We will first consider the special case $P ( x ) = x ^ { k }$, where $k \in \mathbb { N }$.

QII.E.3 Binomial Theorem (positive integer n) Multi-Part Structured Problem Involving Binomial Expansions View

Let $g : [ 0,1 ] \rightarrow \mathbb { R }$ be the function such that $g ( x ) = 1 / x$ if $x \geqslant \mathrm { e } ^ { - 1 }$ and $g ( x ) = 0$ otherwise. We fix a real $\varepsilon \in ] 0 , \mathrm { e } ^ { - 1 } [$. We define two continuous applications $g ^ { + } , g ^ { - } : [ 0,1 ] \rightarrow \mathbb { R }$ as follows:

$g ^ { + }$ is affine on $\left[ \mathrm { e } ^ { - 1 } - \varepsilon , \mathrm { e } ^ { - 1 } \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } - \varepsilon \right] \cup \left[ \mathrm { e } ^ { - 1 } , 1 \right]$;
$g ^ { - }$ is affine on $\left[ \mathrm { e } ^ { - 1 } , \mathrm { e } ^ { - 1 } + \varepsilon \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } \left[ \cup \left[ \mathrm { e } ^ { - 1 } + \varepsilon , 1 \right] \right. \right.$.

Establish the existence of two polynomials $P , Q$ with real coefficients such that: $$\forall x \in [ 0,1 ] , \quad g ^ { - } ( x ) - \varepsilon \leqslant P ( x ) \leqslant g ( x ) \leqslant Q ( x ) \leqslant g ^ { + } ( x ) + \varepsilon$$

QII.E.4 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. Let $g$, $g^+$, $g^-$, $P$, $Q$ be as defined in II.E.1--II.E.3. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$.
Establish the existence of an integer $N _ { 1 } > 0$ such that for every integer $N \geqslant N _ { 1 }$, $$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } P \left( x _ { N } ^ { n } \right) \geqslant \int _ { 0 } ^ { 1 } P ( t ) \mathrm { d } t - \varepsilon$$ and $$\left( 1 - x _ { N } \right) \sum _ { n = 0 } ^ { + \infty } a _ { n } x _ { N } ^ { n } Q \left( x _ { N } ^ { n } \right) \leqslant \int _ { 0 } ^ { 1 } Q ( t ) \mathrm { d } t + \varepsilon$$

QII.E.5 Sequences and Series Proof of Inequalities Involving Series or Sequence Terms View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $A_N = \sum_{k=0}^N a_k$. For every integer $N > 0$ we set $x _ { N } = \mathrm { e } ^ { - 1 / N }$. Let $N_1$ be as in II.E.4.
Deduce from the three previous questions that for every integer $N \geqslant N _ { 1 }$ $$1 - 5 \varepsilon \leqslant \left( 1 - x _ { N } \right) A _ { N } \leqslant 1 + 5 \varepsilon$$

QII.E.6 Sequences and series, recurrence and convergence Coefficient and growth rate estimation View

Let $\left( a _ { n } \right) _ { n \geqslant 0 }$ be a real sequence with $\sum a_n x^n$ having radius of convergence 1, with sum $f(x) = \sum_{n=0}^{+\infty} a_n x^n$ satisfying $f(x) \sim \frac{1}{1-x}$ as $x \to 1^-$, and with $a_n \geqslant 0$ for all $n \in \mathbb{N}$. We denote $\widetilde{a}_n = \frac{A_n}{n+1}$ where $A_n = \sum_{k=0}^n a_k$.
Conclude (i.e., prove property II.3: $\lim_{n \to \infty} \widetilde{a}_n = 1$).

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