The Maclaurin series for the function $f$ is given by $f ( x ) = \sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { k + 1 } x ^ { k } } { k ^ { 2 } } = x - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 3 } } { 9 } - \cdots$ on its interval of convergence.
(a) Use the ratio test to determine the interval of convergence of the Maclaurin series for $f$. Show the work that leads to your answer.
(b) The Maclaurin series for $f$ evaluated at $x = \frac { 1 } { 4 }$ is an alternating series whose terms decrease in absolute value to 0. The approximation for $f \left( \frac { 1 } { 4 } \right)$ using the first two nonzero terms of this series is $\frac { 15 } { 64 }$. Show that this approximation differs from $f \left( \frac { 1 } { 4 } \right)$ by less than $\frac { 1 } { 500 }$.
(c) Let $h$ be the function defined by $h ( x ) = \int _ { 0 } ^ { x } f ( t ) \, d t$. Write the first three nonzero terms and the general term of the Maclaurin series for $h$.

The Maclaurin series for the function $f$ is given by $$f ( x ) = \sum _ { n = 0 } ^ { \infty } \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } = 2 x + \frac { 4 x ^ { 2 } } { 2 } + \frac { 8 x ^ { 3 } } { 3 } + \frac { 16 x ^ { 4 } } { 4 } + \cdots + \frac { ( 2 x ) ^ { n + 1 } } { n + 1 } + \cdots$$ on its interval of convergence.
(a) Find the interval of convergence of the Maclaurin series for $f$. Justify your answer.
(b) Find the first four terms and the general term for the Maclaurin series for $f ^ { \prime } ( x )$.
(c) Use the Maclaurin series you found in part (b) to find the value of $f ^ { \prime } \left( - \frac { 1 } { 3 } \right)$.

The function $f$ has a Taylor series about $x = 2$ that converges to $f(x)$ for all $x$ in the interval of convergence. The $n$th derivative of $f$ at $x = 2$ is given by $f^{(n)}(2) = \frac{(n+1)!}{3^n}$ for $n \geq 1$, and $f(2) = 1$.
(a) Write the first four terms and the general term of the Taylor series for $f$ about $x = 2$.
(b) Find the radius of convergence for the Taylor series for $f$ about $x = 2$. Show the work that leads to your answer.
(c) Let $g$ be a function satisfying $g(2) = 3$ and $g'(x) = f(x)$ for all $x$. Write the first four terms and the general term of the Taylor series for $g$ about $x = 2$.
(d) Does the Taylor series for $g$ as defined in part (c) converge at $x = -2$? Give a reason for your answer.

Let $f$ be a function with derivatives of all orders and for which $f ( 2 ) = 7$. When $n$ is odd, the $n$th derivative of $f$ at $x = 2$ is 0. When $n$ is even and $n \geq 2$, the $n$th derivative of $f$ at $x = 2$ is given by $f ^ { ( n ) } ( 2 ) = \frac { ( n - 1 ) ! } { 3 ^ { n } }$.
(a) Write the sixth-degree Taylor polynomial for $f$ about $x = 2$.
(b) In the Taylor series for $f$ about $x = 2$, what is the coefficient of $( x - 2 ) ^ { 2 n }$ for $n \geq 1$ ?
(c) Find the interval of convergence of the Taylor series for $f$ about $x = 2$. Show the work that leads to your answer.

The function $g$ has derivatives of all orders, and the Maclaurin series for $g$ is
$$\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \frac { x ^ { 2 n + 1 } } { 2 n + 3 } = \frac { x } { 3 } - \frac { x ^ { 3 } } { 5 } + \frac { x ^ { 5 } } { 7 } - \cdots$$
(a) Using the ratio test, determine the interval of convergence of the Maclaurin series for $g$.
(b) The Maclaurin series for $g$ evaluated at $x = \frac { 1 } { 2 }$ is an alternating series whose terms decrease in absolute value to 0. The approximation for $g \left( \frac { 1 } { 2 } \right)$ using the first two nonzero terms of this series is $\frac { 17 } { 120 }$. Show that this approximation differs from $g \left( \frac { 1 } { 2 } \right)$ by less than $\frac { 1 } { 200 }$.
(c) Write the first three nonzero terms and the general term of the Maclaurin series for $g ^ { \prime } ( x )$.

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 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.

Using the fact that $f(x) = g(x)$ on $I = ]-\pi/2, \pi/2[$ where $f(x) = \frac{\sin x + 1}{\cos x}$ and $g$ is the sum of the power series $\sum_{n \in \mathbb{N}} \frac{\alpha_n}{n!} x^n$, deduce that $R = \pi/2$.

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 $r \in [0, \gamma r_0]$.

Let $\sum _ { k \geqslant 0 } c _ { k } x ^ { k }$ be a power series with radius of convergence $R > 0$ and $r \in ] 0 , R [$. Show that there exists $C \in \mathbb { R }$ such that $$\forall k \in \mathbb { N } , \quad \left| c _ { k } \right| \leqslant \frac { C } { r ^ { k } }.$$