The function $g$ has derivatives of all orders for all real numbers. The Maclaurin series for $g$ is given by $g ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } x ^ { n } } { 2 e ^ { n } + 3 }$ on its interval of convergence.
(a) State the conditions necessary to use the integral test to determine convergence of the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$. Use the integral test to show that $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ converges.
(b) Use the limit comparison test with the series $\sum _ { n = 0 } ^ { \infty } \frac { 1 } { e ^ { n } }$ to show that the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ converges absolutely.
(c) Determine the radius of convergence of the Maclaurin series for $g$.
(d) The first two terms of the series $g ( 1 ) = \sum _ { n = 0 } ^ { \infty } \frac { ( - 1 ) ^ { n } } { 2 e ^ { n } + 3 }$ are used to approximate $g ( 1 )$. Use the alternating series error bound to determine an upper bound on the error of the approximation.

The function $f$ is defined by the power series $f ( x ) = x - \frac { x ^ { 3 } } { 3 } + \frac { x ^ { 5 } } { 5 } - \frac { x ^ { 7 } } { 7 } + \cdots + \frac { ( - 1 ) ^ { n } x ^ { 2 n + 1 } } { 2 n + 1 } + \cdots$ for all real numbers $x$ for which the series converges.
(a) Using the ratio test, find the interval of convergence of the power series for $f$. Justify your answer.
(b) Show that $\left| f \left( \frac { 1 } { 2 } \right) - \frac { 1 } { 2 } \right| < \frac { 1 } { 10 }$. Justify your answer.
(c) Write the first four nonzero terms and the general term for an infinite series that represents $f ^ { \prime } ( x )$.
(d) Use the result from part (c) to find the value of $f ^ { \prime } \left( \frac { 1 } { 6 } \right)$.

The function $f$ has derivatives of all orders for all real numbers. It is known that $f(0) = 2$, $f'(0) = 3$, $f''(x) = -f\left(x^{2}\right)$, and $f'''(x) = -2x \cdot f'\left(x^{2}\right)$.
(a) Find $f^{(4)}(x)$, the fourth derivative of $f$ with respect to $x$. Write the fourth-degree Taylor polynomial for $f$ about $x = 0$. Show the work that leads to your answer.
(b) The fourth-degree Taylor polynomial for $f$ about $x = 0$ is used to approximate $f(0.1)$. Given that $\left|f^{(5)}(x)\right| \leq 15$ for $0 \leq x \leq 0.5$, use the Lagrange error bound to show that this approximation is within $\frac{1}{10^{5}}$ of the exact value of $f(0.1)$.
(c) Let $g$ be the function such that $g(0) = 4$ and $g'(x) = e^{x} f(x)$. Write the second-degree Taylor polynomial for $g$ about $x = 0$.

The Taylor series for a function $f$ about $x = 4$ is given by $$\sum _ { n = 1 } ^ { \infty } \frac { ( x - 4 ) ^ { n + 1 } } { ( n + 1 ) 3 ^ { n } } = \frac { ( x - 4 ) ^ { 2 } } { 2 \cdot 3 } + \frac { ( x - 4 ) ^ { 3 } } { 3 \cdot 3 ^ { 2 } } + \frac { ( x - 4 ) ^ { 4 } } { 4 \cdot 3 ^ { 3 } } + \cdots + \frac { ( x - 4 ) ^ { n + 1 } } { ( n + 1 ) 3 ^ { n } } + \cdots$$ and converges to $f ( x )$ on its interval of convergence.
A. Using the ratio test, find the interval of convergence of the Taylor series for $f$ about $x = 4$. Justify your answer.
B. Find the first three nonzero terms and the general term of the Taylor series for $f ^ { \prime }$, the derivative of $f$, about $x = 4$.
C. The Taylor series for $f ^ { \prime }$ described in part B is a geometric series. For all $x$ in the interval of convergence of the Taylor series for $f ^ { \prime }$, show that $f ^ { \prime } ( x ) = \frac { x - 4 } { 7 - x }$.
D. It is known that the radius of convergence of the Taylor series for $f$ about $x = 4$ is the same as the radius of convergence of the Taylor series for $f ^ { \prime }$ about $x = 4$. Does the Taylor series for $f ^ { \prime }$ described in part B converge to $f ^ { \prime } ( x ) = \frac { x - 4 } { 7 - x }$ at $x = 8$ ? Give a reason for your answer.

The power series $$\sum_{n=1}^{\infty} \frac{n^2 x^n}{n!}$$ equals
(A) $x^2 e^x$;
(B) $x e^x$;
(C) $(x^2 + x) e^x$;
(D) $(x^2 - x) e^x$;

Find the Taylor series expansion of $f ( x ) = e ^ { x } \sin ( x )$ centered at $x = \frac { \pi } { 4 }$.

For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
Deduce from the above that $F_n$ extends to $\mathbb{R}$ as a unique polynomial function, whose degree and leading coefficient should be specified.

Throughout the rest of this problem, we set $T_0(x) = 1$. For $n \in \mathbb{N}^*$, we denote by $T_n$ the polynomial function satisfying $T_n(x) = 2^{1-n} F_n(x)$ for all $x \in \mathbb{R}$.
Determine two real numbers $a$ and $b$ such that $$\forall x \in \mathbb{R}, \forall n \in \mathbb{N}^*, T_{n+2}(x) = a x T_{n+1}(x) + b T_n(x)$$

We study the differential equation $$y(x) y'(x) = -4x \tag{E}$$
Let $m$ be the maximal solution determined in question II.B.3).
II.D.1) Show that the solution $m$ is expandable as a power series in a neighborhood of 0. Calculate this expansion and specify its radius of convergence. II.D.2) Deduce the power series expansions of all maximal solutions of $(E)$; specify the radii of convergence of these power series.

Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$. By applying to $f$ the Taylor formula with integral remainder at order 2, show that for every $k \in \mathbb { N } ^ { * } , f ( k + 1 ) - f ( k ) = \frac { 1 } { k ^ { \alpha } } - \frac { \alpha } { 2 } \frac { 1 } { k ^ { \alpha + 1 } } + A _ { k }$ where $A _ { k }$ is a real number satisfying $0 \leqslant A _ { k } \leqslant \frac { \alpha ( \alpha + 1 ) } { 2 k ^ { \alpha + 2 } }$.

Let $f$ be the function defined on $\mathbb { R } _ { + } ^ { * }$ by $f ( x ) = \frac { 1 } { ( 1 - \alpha ) x ^ { \alpha - 1 } }$, where $\alpha$ is a real number strictly greater than 1. We fix a non-zero natural integer $p$ and denote by $g = a _ { 0 } f + a _ { 1 } f ^ { \prime } + \cdots + a _ { 2 p - 1 } f ^ { ( 2 p - 1 ) }$. For every $k \in \mathbb { N } ^ { * }$, we set $R ( k ) = g ( k + 1 ) - g ( k ) - f ^ { \prime } ( k )$ so that $g ( k + 1 ) - g ( k ) = f ^ { \prime } ( k ) + R ( k )$.
By applying to $g$ the Taylor formula with integral remainder at order $2 p$, show that there exists a real number $A$ such that for every $k \in \mathbb { N } ^ { * } , | R ( k ) | \leqslant A k ^ { - ( 2 p + \alpha ) }$.

Let $g$ denote a function of class $C^1$ from $\mathbb{R}^n$ to $\mathbb{R}$. We fix an element $a = (a_1, a_2, \ldots, a_n)$ of $\mathbb{R}^n$. Let $\varphi$ be the function from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\varphi(t) = g(ta) = g(ta_1, ta_2, \ldots, ta_n)$$ Deduce that in a neighbourhood of 0 $$g(ta) = g(0) + t\left(a_1 \mathrm{D}_1 g(0) + a_2 \mathrm{D}_2 g(0) + \cdots + a_n \mathrm{D}_n g(0)\right) + \mathrm{o}(t)$$

Let $z \in \mathbb{C}$. Consider the function of the real variable $x$ $$G_z : x \mapsto \sum_{p=0}^{+\infty} \left(x^p(2z - x)^p\right)$$ Deduce (from II.C.5) that $G_z$ admits a Taylor expansion to any order at 0. We denote it $$G_z(x) = \sum_{k=0}^{n} a_k x^k + o\left(x^n\right) \quad x \to 0$$ Determine the coefficients $a_k$ for $k \in \mathbb{N}$.

We recall that $\phi$ is defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$, and that two power series $\sum _ { k \geqslant 1 } b _ { k } q ^ { k }$ and $\sum _ { k \geqslant 1 } c _ { k } q ^ { k }$ with strictly positive radius of convergence, $b_1 > 0$, $c_1 < 0$, satisfy $\phi \left( \sum _ { k = 1 } ^ { \infty } b _ { k } q ^ { k } \right) = \phi \left( \sum _ { k = 1 } ^ { \infty } c _ { k } q ^ { k } \right) = q ^ { 2 }$ for $q$ near 0 in $[0,+\infty[$.
Calculate $b _ { 1 } , b _ { 2 } , b _ { 3 }$ and $c _ { 1 } , c _ { 2 }$ and $c _ { 3 }$. Deduce the following asymptotic expansions when $q \rightarrow 0 , q > 0$, for the functions $\phi _ { - } ^ { - 1 }$ and $\phi _ { + } ^ { - 1 }$ as well as their derivatives: $$\begin{array} { l l } \phi _ { + } ^ { - 1 } ( q ) = \sqrt { 2 q } + \frac { 2 q } { 3 } + \frac { q ^ { 3 / 2 } } { 9 \sqrt { 2 } } + o \left( q ^ { 3 / 2 } \right) , & \phi _ { - } ^ { - 1 } ( q ) = - \sqrt { 2 q } + \frac { 2 q } { 3 } - \frac { q ^ { 3 / 2 } } { 9 \sqrt { 2 } } + o \left( q ^ { 3 / 2 } \right) \\ \left( \phi _ { + } ^ { - 1 } \right) ^ { \prime } ( q ) = \frac { 1 } { \sqrt { 2 q } } + \frac { 2 } { 3 } + \frac { \sqrt { q } } { 6 \sqrt { 2 } } + o ( \sqrt { q } ) , & \left( \phi _ { - } ^ { - 1 } \right) ^ { \prime } ( q ) = - \frac { 1 } { \sqrt { 2 q } } + \frac { 2 } { 3 } - \frac { \sqrt { q } } { 6 \sqrt { 2 } } + o ( \sqrt { q } ) \end{array}$$

Using the results of the previous questions, deduce that $$\Gamma ( y ) = e ^ { - y } y ^ { y } \left( \frac { 2 \pi } { y } \right) ^ { 1 / 2 } \left( 1 + \frac { 1 } { 12 y } + o \left( \frac { 1 } { y } \right) \right) \quad \text { when } y \rightarrow + \infty .$$

We assume that $m = 0$ and there exists $\lambda > 0$ such that $G(0) = (0,0,2\lambda)$, $G^{\prime}(0) = (1,0,0)$, and $G$ satisfies $$\forall x \in \mathbb{R}, G^{\prime\prime\prime}(x) + \left(\lambda^{2} + \frac{x^{2}}{4}\right) G^{\prime}(x) - \frac{x}{4} G(x) = 0$$
(a) Show that $$\forall x \in \mathbb{R}, \left|G_{1}^{\prime}(x) - \cos(\lambda x)\right| \leq \frac{|x|^{3}}{6\lambda}$$ Hint: One may assume that for $r \in C(\mathbb{R}, \mathbb{R})$, if $y \in C^{2}(\mathbb{R}, \mathbb{R})$ satisfies $y^{\prime\prime}(x) + \lambda^{2} y(x) = r(x)$ for all $x \in \mathbb{R}$, then $$y(x) = \cos(\lambda x)\, y(0) + \frac{\sin(\lambda x)}{\lambda}\, y^{\prime}(0) + \frac{1}{\lambda}\int_{0}^{x} r(s)\sin(\lambda(x-s))\,ds$$
(b) Deduce that there exists $\lambda_{0} > 0$ such that if $\lambda > \lambda_{0}$ then there exists $x_{0} \neq 0$ such that $G_{1}(x_{0}) = 0$.

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Show that in a neighbourhood of $x = 0$, the function $F$ can be written in the form
$$F(x) = \sum_{n=0}^{+\infty} c_{n} \frac{(\mathrm{i}x)^{n}}{n!} \tag{S}$$
where $c_{n}$ is the value of Gamma at a point to be specified. Express $c_{n}$ in terms of $n$ and $c_{0}$.
What is the radius of convergence of the power series appearing on the right-hand side of $(S)$?

For $x \in \mathbb{R}$, we define $F(x) = \int_{0}^{+\infty} \mathrm{e}^{-t} t^{-3/4} \mathrm{e}^{\mathrm{i}tx} \mathrm{~d}t$. Let $R(x)$ be the real part and $I(x)$ be the imaginary part of $F(x)$.
Determine, in a neighbourhood of 0, the Taylor expansion of $R(x)$ to order 3 and of $I(x)$ to order 4.

We consider the function $\psi$ defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}^{*}, \quad \psi(x) = \frac{\sin(\pi x)}{\pi x} \quad \text{and} \quad \psi(0) = 1$$
I.B.1) Justify that $\psi$ is expandable as a power series. Specify this expansion and its radius of convergence. Deduce that $\psi$ is of class $C^{\infty}$ on $\mathbb{R}$.
I.B.2) Prove that
$$\forall n \in \mathbb{N}, \quad \int_{n}^{n+1} |\psi(x)| \mathrm{d}x \geqslant \frac{2}{(n+1)\pi^{2}}$$
Deduce that $\psi$ does not belong to $E_{\mathrm{cpm}}$.

Let $f \in \mathcal{S}$.
I.D.1) Justify that, for every natural number $n$, the function $x \mapsto x^{n} f(x)$ is integrable on $\mathbb{R}$.
I.D.2) Prove that the function $\mathcal{F}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that
$$\forall n \in \mathbb{N}, \quad \forall \xi \in \mathbb{R}, \quad (\mathcal{F}(f))^{(n)}(\xi) = (-2\pi\mathrm{i})^{n} \int_{-\infty}^{+\infty} t^{n} f(t) e^{-2\pi\mathrm{i} t\xi} \mathrm{~d}t$$

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
Prove that $\mathcal{F}(f)$ is of class $C^{\infty}$ on $\mathbb{R}$ and that $\mathcal{F}(f) \in \mathcal{S}$. Deduce that $f$ is of class $C^{\infty}$ on $\mathbb{R}$.

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
Prove that
$$\forall (x, x_{0}) \in \mathbb{R}^{2}, \quad \sum_{n=0}^{+\infty} \frac{(x-x_{0})^{n}}{n!} \int_{-1/2}^{1/2} (2\pi\mathrm{i}\xi)^{n} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x_{0}\xi} \mathrm{d}\xi = f(x)$$

Let $f$ be a function in $\mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. According to the Fourier inversion formula, we have
$$\forall x \in \mathbb{R}, \quad f(x) = \int_{-1/2}^{1/2} \mathcal{F}(f)(\xi) e^{2\pi\mathrm{i} x\xi} \mathrm{d}\xi$$
We assume that $f$ is zero outside a segment $[a, b]$. Show that $f = 0$.

Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. The sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ is defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Prove the existence of a real number $E$ such that
$$\forall t \in \left[-\frac{1}{2}, \frac{1}{2}\right], \quad \left|f(t) - \sum_{k=-n}^{n} c_{k}(f) e^{2\pi\mathrm{i} kt}\right| \leqslant \frac{E}{2n+1}$$
One may introduce the function $h_{t} : x \mapsto f(x+t)$.

Let $f \in \mathcal{S}$ whose Fourier transform $\mathcal{F}(f)$ is zero outside the segment $[-1/2, 1/2]$. We set
$$\forall k \in \mathbb{Z}, \quad \forall x \in \mathbb{R}, \quad \psi_{k}(x) = \psi(x+k)$$
where $\psi(x) = \frac{\sin(\pi x)}{\pi x}$ for $x \neq 0$ and $\psi(0) = 1$.
Justify that $\forall n \in \mathbb{N}, \quad (\mathcal{F}(f))^{(n)}\left(\frac{1}{2}\right) = (\mathcal{F}(f))^{(n)}\left(-\frac{1}{2}\right) = 0$.