grandes-ecoles 2013 QIV.C.5

grandes-ecoles · France · centrale-maths2__mp Sequences and Series Asymptotic Equivalents and Growth Estimates for Sequences/Series
In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $M_n = \sum_{k=1}^n \dfrac{1}{2\cos\dfrac{k\pi}{2n+1}}$.
Give an equivalent of $M_n$ as $n$ tends to $+\infty$. 
In this question, $f$ denotes the linear form defined by $\forall M \in \mathcal{M}_n(\mathbb{R}), f(M) = \sum_{j=1}^n \sum_{i=j}^n m_{i,j}$, and $M_n = \sum_{k=1}^n \dfrac{1}{2\cos\dfrac{k\pi}{2n+1}}$.

Give an equivalent of $M_n$ as $n$ tends to $+\infty$.
Paper Questions
QI.A.1 QI.A.2 QI.B.1 QI.B.2 QI.B.3 QI.C.1 QI.C.2 QI.C.3 QI.D.1 QI.D.2 QI.D.3 QI.D.4 QI.E QII.A.1 QII.A.2 QII.A.3 QII.B.1 QII.B.2 QIII.A QIII.B QIII.C QIII.D QIV.A QIV.B.1 QIV.B.2 QIV.B.3 QIV.B.4 QIV.C.1 QIV.C.2 QIV.C.3 QIV.C.4 QIV.C.5