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For b/

with the relation in a/

a^(k+1)/(x(k+1)-a)= a^k/(x(k)-a) - a^k/x(k)

a) ok by induction

Can someone post the solution of b) please ?

thanks

a/2

1/(k+1)

Here a version for functions of class \( C^{n+1} \):

Let \( a>0 \) and f a function from [0;a] in R, \( f\in C^{n+1} \),
with \( f(0)=f^{\prime} (0)=...=f^{(n)}(0)=0 \).

Prove that

\( \int_{0}^{a}|ff^{(n+1)}|\leq \frac{a^{n+1}}{n!\sqrt{(2n+1)(2n+2)}}\int_{0}^{a}(f^{(n+1)})^2 \).

\( x(0)>0 \)

\( x(n+1)=x(n) - e^{\frac{-1}{x(n)^2}} \)

Prove that \( x(n)=\frac{1}{\sqrt{\ln n}} - \frac{3}{4}\frac{\ln(\ln n)}{(\ln n)^{3/2}} + \frac{u(n)\ln(\ln n)}{(\ln n)^{3/2}} \)

with u(n) tend to 0 when n tend to infinity.

Let \( F=(\sin p\ ;\ p\in P) \), where \( P \) is the set of prime numbers.

Does F is dense in \( [-1;1] \) ?

x(1)>0

\( x(n+1)=\frac{x(n)}{1+n{x(n)}^2} \)

1/ Study the convergence of x(n).

2/ Find an equivalent of x(n).

a/ Solve in \( M(3,C) \) the equation \( X^2=A \)
where

A=
(9 0 0)
(0 4 0)
(-1 0 1)

b/ Solve the equation \( X^2=B \)

in \( M(3n,C) \) with \( n \geq 1 \)

and \( B=diag(A,...,A) \) here A appears n-times

Prove that for any matrix \( A\in M(n,K) \), with \( K= \mathbb R\ \text{or}\ \mathbb C \), there exist a sequence of invertible matrices \( A_{q}\in GL(n,K) \) such that \( A_{q} \) tend to \( A \) when \( q \) tend to \( \infty \).

Let f be a real function defined on [0;1] which is \( C^3 \) (it means that f''' there exists and it is continous).

Find a, b, c such that

\( \frac{1}{n}\sum_{k=1}^{n}f(k/n)=a+\frac{b}{n}+\frac{c}{n^2}+\frac{u(n)}{n^2} \)

with u(n) tend to zero when n tend \( \infty \).

Let f be a continous function from R^2 \rightarrow R and (b_{n}),\ (c_{n}) real sequences such that f(b_{n}) tend to \infty when n tend to \infty and f(c_{n}) tend to -\infty when n tend to \infty . Prove that there exists (a_{n}) a real sequence such that f(a_{n})=0 for any n\in N and ||a_{n}|| ten...

Let E bet the set of matrices A in M(2n,R) such that a_{ii}=0 for any 1\leq i\leq 2n a_{ij} is -1 or 1 for any 1\leq i\neq j\leq 2n 1/ For A in E prove det(A) is an odd integer. 2/ Find the maximum of |\det(A)| for any matrix A \in E and find the minimum of |\det(A)| for any A \in E .

\det (A \overline A+I_{n}) se scrie (\alpha_1 \overline \alpha_1 +1)(\alpha_2 \overline \alpha_2 +1)...(\alpha_n \overline \alpha_n +1) Take A= (1 1) (-1 1) with eigenvalues 1+i, 1-i A\overline A=A^2= (0 2) (-2 0) with eigenvalues 2i,-2i \det(A\overline A+I)=\det(A^2+I)=5 On the other side ((1+i)(1...

Is it possible to have the problems of Joseph Wildt 2008 ?

Beniamin you got the right idea
-------------------------------------

Sorry there is small mistake a(3,3) is -1 not 1

the matrix is

M=
(0 0 -1)
(0 1 1)
(1 1 -1)