The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { n = 0 } ^ { \infty } \left( - \frac { x } { 4 } \right) ^ { n }$. What is the value of $f ( 3 )$ ?
(A) - 3
(B) $- \frac { 3 } { 7 }$
(C) $\frac { 4 } { 7 }$
(D) $\frac { 13 } { 16 }$
(E) 4

For $x > 0$, the power series $1 - \frac { x ^ { 2 } } { 3 ! } + \frac { x ^ { 4 } } { 5 ! } - \frac { x ^ { 6 } } { 7 ! } + \cdots + ( - 1 ) ^ { n } \frac { x ^ { 2 n } } { ( 2 n + 1 ) ! } + \cdots$ converges to which of the following?
(A) $\cos x$
(B) $\sin x$
(C) $\frac { \sin x } { x }$
(D) $e ^ { x } - e ^ { x ^ { 2 } }$
(E) $1 + e ^ { x } - e ^ { x ^ { 2 } }$

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) $(x^2 - x) e^x$;

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 $v _ { n }$.

Using question 15, determine, for any real $\xi$, the value of $K(\xi)$.

Recall, without justification, the expression of $t^{\prime}$ as a function of $t$, where $t(x) = \tan(x)$.

Show that $$\forall x \in ]-1,1[ \quad s(x)e^{x} = \frac{1}{1-x}$$ Deduce that $R = 1$.

If $0 < a , b < 1$, and $\tan ^ { - 1 } a + \tan ^ { - 1 } b = \frac { \pi } { 4 }$, then the value of $( a + b ) - \left( \frac { a ^ { 2 } + b ^ { 2 } } { 2 } \right) + \left( \frac { a ^ { 3 } + b ^ { 3 } } { 3 } \right) - \left( \frac { a ^ { 4 } + b ^ { 4 } } { 4 } \right) + \ldots$ is :
(1) $\log _ { \mathrm { e } } \left( \frac { e } { 2 } \right)$
(2) $e$
(3) $e ^ { 2 } - 1$
(4) $\log _ { e } 2$