SEEMOUS 2023

Prove that if $A{}$ and $B{}$ are $n\times n$ matrices with complex entries which satisfy $$A=AB-BA+A^2B-2ABA+BA^2+A^2BA-ABA^2,$$then $\det(A)=0$ .

For the sequence $$S_n=\frac{1}{\sqrt{n^2+1^2}}+\frac{1}{\sqrt{n^2+2^2}}+\cdots+\frac{1}{\sqrt{n^2+n^2}},$$find the limit $$\lim_{n\to\infty}n\left(n\cdot\left(\log(1+\sqrt{2})-S_n\right)-\frac{1}{2\sqrt{2}(1+\sqrt{2})}\right).$$

Prove that if $A{}$ is an $n\times n$ matrix with complex entries such that $A+A^*=A^2A^*$ then $A=A^*$ . (Here, we denote by $M^*$ the conjugate transpose $\overline{M}^t$ of the matrix $M{}$ ).

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous, strictly decreasing function such that $f([0,1])\subseteq[0,1]$ . [list=i] []For all positive integers $n{}$ prove that there exists a unique $a_n\in(0,1)$ , solution of the equation $f(x)=x^n$ . Moreover, if $(a_n){}$ is the sequence defined as above, prove that $\lim_{n\to\infty}a_n=1$ . []Suppose $f$ has a continuous derivative, with $f(1)=0$ and $f'(1)<0$ . For any $x\in\mathbb{R}$ we define $$F(x)=\int_x^1f(t) \ dt.$$Let $\alpha{}$ be a real number. Study the convergence of the series $$\sum_{n=1}^\infty F(a_n)^\alpha.$$ [/list]