UFM Pure

grandes-ecoles 2021 Q18b Formal power series manipulation (Cauchy product, algebraic identities) View

Using the result that for all $x \in ]0,1[$: $$\frac{\pi}{\sin(\pi x)} = \sum_{n=0}^{+\infty} \frac{(-1)^n}{n+x} + \sum_{n=0}^{+\infty} \frac{(-1)^n}{n+1-x},$$ deduce that, for $x \in ]-\frac{1}{2}, \frac{1}{2}[$: $$\frac{\pi}{\cos(\pi x)} = \sum_{k=0}^{+\infty} \left(\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^{2k+1}}\right) 2^{2k+2} x^{2k}.$$

grandes-ecoles 2021 Q18c Prove smoothness or power series expandability of a function View

Using the result that for $x \in ]-\frac{1}{2}, \frac{1}{2}[$: $$\frac{\pi}{\cos(\pi x)} = \sum_{k=0}^{+\infty} \left(\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^{2k+1}}\right) 2^{2k+2} x^{2k},$$ deduce that the function $$v : \begin{array}{ccc} ]-\frac{\pi}{2}, \frac{\pi}{2}[ & \longrightarrow & \mathbb{R} \\ x & \longmapsto & \frac{1}{\cos(x)} \end{array}$$ is expandable as a power series and that, for all $k \in \mathbb{N}$, $$\sum_{n=0}^{+\infty} \frac{(-1)^n}{(2n+1)^{2k+1}} = \frac{\pi^{2k+1}}{2^{2k+2}(2k)!} E_{2k}$$ where, for all $k \in \mathbb{N}$, $E_{2k} = v^{(2k)}(0)$.

grandes-ecoles 2021 Q24 Construct Taylor/Maclaurin polynomial from derivative values View

Let $n \in \mathbb{N}$. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Determine $L_0, L_1, L_2$ and $L_3$.

grandes-ecoles 2021 Q25 Formal power series manipulation (Cauchy product, algebraic identities) View

Let $n \in \mathbb{N}$ be a non-zero natural integer. We define, for any real number $x$, $$\Phi_n(x) = \mathrm{e}^{-x} x^n \quad \text{and} \quad L_n(x) = \frac{\mathrm{e}^x}{n!} \Phi_n^{(n)}(x).$$ Using Leibniz's formula, prove that the function $L_n$ is polynomial of degree $n$. Determine the coefficients $c_{n,k}$ such that $L_n(x) = \sum_{k=0}^{n} c_{n,k} x^k$.

grandes-ecoles 2022 Q5a Construct series for a composite or related function View

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Show that: $$\forall x \in ]-2\pi, 2\pi[ \backslash \{0\}, \quad \frac{x}{2} \operatorname{cotan}\left(\frac{x}{2}\right) = 1 - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} x^{2k}.$$

grandes-ecoles 2022 Q5b Construct series for a composite or related function View

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Deduce: $$\forall x \in ]-2\pi, 2\pi[ \backslash \{0\}, \quad \frac{ix}{e^{ix}-1} = 1 - \frac{ix}{2} - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} \cdot x^{2k}.$$

grandes-ecoles 2022 Q5a Construct series for a composite or related function View

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Using the identity $\pi x \operatorname{cotan}(\pi x) = 1 + 2\sum_{n=1}^{+\infty} \frac{x^2}{x^2 - n^2}$, show that: $$\forall x \in \left]-2\pi, 2\pi\right[ \backslash \{0\}, \quad \frac{x}{2}\operatorname{cotan}\left(\frac{x}{2}\right) = 1 - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} x^{2k}$$

grandes-ecoles 2022 Q5b Construct series for a composite or related function View

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Using the result $$\forall x \in \left]-2\pi, 2\pi\right[ \backslash \{0\}, \quad \frac{x}{2}\operatorname{cotan}\left(\frac{x}{2}\right) = 1 - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} x^{2k}$$ deduce: $$\forall x \in \left]-2\pi, 2\pi\right[ \backslash \{0\}, \quad \frac{ix}{e^{ix}-1} = 1 - \frac{ix}{2} - \sum_{k=1}^{+\infty} \frac{\zeta(2k)}{2^{2k-1}\pi^{2k}} \cdot x^{2k}$$

grandes-ecoles 2022 Q6 Formal power series manipulation (Cauchy product, algebraic identities) View

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
Show that for all $z \in \mathbb{C}$ such that $|z| < 2\pi$, we have $$z = \left(e^z - 1\right)\left(1 - \frac{z}{2} + \sum_{k=1}^{+\infty} \frac{(-1)^{k-1} \zeta(2k)}{2^{2k-1}\pi^{2k}} z^{2k}\right).$$

grandes-ecoles 2022 Q6 Formal power series manipulation (Cauchy product, algebraic identities) View

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Show that for all $z \in \mathbb{C}$ such that $|z| < 2\pi$, we have $$z = \left(e^z - 1\right)\left(1 - \frac{z}{2} + \sum_{k=1}^{+\infty} \frac{(-1)^{k-1}\zeta(2k)}{2^{2k-1}\pi^{2k}} z^{2k}\right)$$

grandes-ecoles 2022 Q7a Extract derivative values from a given series View

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$.
Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$
Show that the function $h$ is of class $C^\infty$ on $\mathbb{R}$ and that, for all $n \in \mathbb{N}^*$, we have $$h^{(2n)}(0) = \frac{(-1)^{n-1}(2n)!}{\pi^{2n} 2^{2n-1}} \zeta(2n).$$

grandes-ecoles 2022 Q7a Extract derivative values from a given series View

For every integer $k \geqslant 2$, we set $\zeta(k) = \sum_{n=1}^{+\infty} n^{-k}$. Let $h$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad h(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Show that the function $h$ is of class $C^\infty$ on $\mathbb{R}$ and that, for all $n \in \mathbb{N}^*$, we have $$h^{(2n)}(0) = \frac{(-1)^{n-1}(2n)!}{\pi^{2n} 2^{2n-1}} \zeta(2n)$$

grandes-ecoles 2022 Q11 Formal power series manipulation (Cauchy product, algebraic identities) View

We consider a power series $g \in O_1$ with non-negative real coefficients. We assume that there exist $a > 0, b > 0$ such that $$g \prec a\left(I + \frac{g^2}{b - g}\right)$$ Show that there exist $r > 0$ and a function $h: ]-r, r[ \longrightarrow \mathbb{R}$, expandable as a power series at 0, satisfying $h(0) = 0$ and such that $$h(x) = a\left(x + \frac{h(x)^2}{b - h(x)}\right)$$ for all $x \in ]-r, r[$. We also denote by $h$ the element of $O_1$ associated with the function $h$.

grandes-ecoles 2022 Q16 Taylor's formula with integral remainder or asymptotic expansion View

Conclude that
$$\ln P \left( e ^ { - t } \right) = \frac { \pi ^ { 2 } } { 6 t } + \frac { \ln ( t ) } { 2 } - \frac { \ln ( 2 \pi ) } { 2 } + o ( 1 ) \quad \text { when } t \text { tends to } 0 ^ { + } .$$

grandes-ecoles 2022 Q16 Taylor's formula with integral remainder or asymptotic expansion View

Conclude that $$\ln P(e^{-t}) = \frac{\pi^2}{6t} + \frac{\ln(t)}{2} - \frac{\ln(2\pi)}{2} + o(1) \text{ when } t \text{ tends to } 0^+.$$

grandes-ecoles 2022 Q18 Lagrange error bound application View

Let $n$ be a nonzero natural integer, $I = [a,b]$ with $a < b$, and $a_1 < \cdots < a_n$ distinct real numbers in $I$. Let $f$ be a real-valued function of class $\mathcal{C}^n$ on $I$ and $P = \Pi(f)$ its Lagrange interpolation polynomial. Deduce that $$\sup _ { x \in [ a , b ] } | f ( x ) - P ( x ) | \leqslant \frac { M _ { n } ( b - a ) ^ { n } } { n ! }$$ where $M _ { n } = \sup _ { x \in [ a , b ] } \left| f ^ { ( n ) } ( x ) \right|$.

grandes-ecoles 2022 Q19 Lagrange error bound application View

Let $I = [ a , b ]$ where $a < b$, and let $f(x) = \exp(x)$ for all $x \in I$. For all $n \in \mathbb { N } ^ { * }$, let $P _ { n } = \Pi _ { n } ( f )$ be the Lagrange interpolation polynomial of $f$ at $n$ distinct points $a_{1,n} < \cdots < a_{n,n}$ of $I$. Show that the sequence $\left( P _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $I$.

grandes-ecoles 2022 Q21 Prove smoothness or power series expandability of a function View

Let $a > 0$, $I = [-a, a]$, and $$f : \begin{array}{ccc} \mathbb{R} & \rightarrow & \mathbb{R} \\ x & \mapsto & \dfrac{1}{1+x^2} \end{array}.$$ Show that $f$ is of class $\mathcal { C } ^ { \infty }$ and that, for all $k$ in $\mathbb { N }$ and all $t \in \left] - \pi / 2 , \pi / 2 \right[$, $$f ^ { ( k ) } ( \tan t ) = k ! \cos ^ { k + 1 } ( t ) \cos ( ( k + 1 ) t + k \pi / 2 ).$$

grandes-ecoles 2022 Q22 Lagrange error bound application View

Let $x \in [ 0,1 [$ and $\theta \in \mathbf { R }$. Using the function $L$, show that
$$\left| \frac { 1 - x } { 1 - x e ^ { i \theta } } \right| \leq \exp ( - ( 1 - \cos \theta ) x ) .$$
Deduce that for all $x \in [ 0,1 [$ and all real $\theta$,
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 1 - x } + \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \right)$$

grandes-ecoles 2022 Q22 Lagrange error bound application View

Let $x \in [0,1[$ and $\theta \in \mathbf{R}$. Using the function $L$, show that $$\left|\frac{1-x}{1-xe^{i\theta}}\right| \leq \exp(-(1-\cos\theta)x)$$ Deduce that for all $x \in [0,1]$ and all real $\theta$, $$\left|\frac{P(xe^{i\theta})}{P(x)}\right| \leq \exp\left(-\frac{1}{1-x} + \operatorname{Re}\left(\frac{1}{1-xe^{i\theta}}\right)\right)$$

grandes-ecoles 2022 Q22 Lagrange error bound application View

Let $a > 0$, $I = [-a, a]$, and $f(x) = \dfrac{1}{1+x^2}$ for $x \in \mathbb{R}$. For all $n \in \mathbb { N } ^ { * }$, let $P _ { n } = \Pi _ { n } ( f )$ be the Lagrange interpolation polynomial of $f$ on $I$. Show that, if $a < \frac { 1 } { 2 }$, the sequence of polynomials $\left( P _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges uniformly towards $f$ on $[ - a , a ]$.

grandes-ecoles 2022 Q23 Lagrange error bound application View

Let $x \in [ 0,1 [$ and $\theta$ a real number. Show that
$$\frac { 1 } { 1 - x } - \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \geq \frac { x ( 1 - \cos \theta ) } { ( 1 - x ) \left( ( 1 - x ) ^ { 2 } + 2 x ( 1 - \cos \theta ) \right) }$$
Deduce that if $x \geq \frac { 1 } { 2 }$ then
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 - \cos \theta } { 6 ( 1 - x ) ^ { 3 } } \right) \quad \text { or } \quad \left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 3 ( 1 - x ) } \right)$$
For this last result, distinguish two cases according to the relative values of $x ( 1 - \cos \theta )$ and $( 1 - x ) ^ { 2 }$.

grandes-ecoles 2022 Q23 Lagrange error bound application View

Let $x \in [0,1[$ and $\theta$ a real. Show that $$\frac{1}{1-x} - \operatorname{Re}\left(\frac{1}{1-xe^{i\theta}}\right) \geq \frac{x(1-\cos\theta)}{(1-x)\left((1-x)^2 + 2x(1-\cos\theta)\right)}.$$ Deduce that if $x \geq \frac{1}{2}$ then $$\left|\frac{P(xe^{i\theta})}{P(x)}\right| \leq \exp\left(-\frac{1-\cos\theta}{6(1-x)^3}\right) \quad \text{or} \quad \left|\frac{P(xe^{i\theta})}{P(x)}\right| \leq \exp\left(-\frac{1}{3(1-x)}\right).$$ For this last result, distinguish two cases according to the relative values of $x(1-\cos\theta)$ and $(1-x)^2$.

grandes-ecoles 2022 Q23 Derive series via differentiation or integration of a known series View

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$. We set $$\forall x \in ] - 1,1 [ , \quad g ( x ) = \sum _ { k = 0 } ^ { + \infty } x ^ { k }.$$ Show that $g$ is of class $\mathcal { C } ^ { \infty }$ on $] - 1,1 [$ and that $$\forall j \in \mathbb { N } , \quad \forall x \in ] - 1,1 [ , \quad g ^ { ( j ) } ( x ) = \frac { j ! } { ( 1 - x ) ^ { j + 1 } }.$$

grandes-ecoles 2022 Q24 Determine radius or interval of convergence View

We now study the linearization problem in the case $|\lambda| = 1$, with $\lambda$ not a root of unity. We set, for $m \geqslant 1$, $$\alpha_m := \min(1/5, \omega_{m+1}, \omega_{m+2}, \ldots, \omega_{2m}), \quad \gamma_m := \alpha_m^{2/m},$$ where $\omega_k := |\lambda^k - \lambda|, k \geqslant 2$.
We are given a series $F \in O_{m+1}, m \geqslant 1$ such that $\rho(F) > 0$. Show that there exists $r_0 \in ]0,1[$ such that $\hat{F}(r) \leqslant r$ for all $r \in [0, r_0]$. Show then, for $\gamma \in ]0,1[$, that $$\hat{F}(r) \leqslant \gamma^m r$$ for all $r \in [0, \gamma r_0]$.

UFM Pure

Sequences and series, recurrence and convergence 375

Roots of polynomials 146

Polar coordinates 22

Conic sections 291

Taylor series 183

Hyperbolic functions 10

Integration using inverse trig and hyperbolic functions 6

Integration with Partial Fractions 7

Vectors: Lines & Planes 293

Vectors: Cross Product & Distances 20

First order differential equations (integrating factor) 60

Complex numbers 2 113

Second order differential equations 127

Systems of differential equations 41