This document discusses the diagonalization of matrices, finding eigenvalues and eigenvectors, and determining the convergence of series.
Extract
[...] So is convergent. 4. so the series it is divergent. so and are the general terms of convergent geometric series. Therefore is convergent. donc . From where . Finally . is the general term of a divergent series. So the series is divergent. 7. here is the general term of a convergent Riemann series. So is convergent. 8. here is the general term of a divergent series. So the series is divergent. is the general term of a convergent Riemann series. [...]
[...] The vectors the component verifies: . We must therefore solve the system of three equations with 3 unknowns as follows: So it: The two equations are equivalent and we therefore obtain a proper subspace of dimension 2 with a basis, for example: [...]
[...] On a where is the diagonal matrix composed of the eigenvalues of . On calculates by the Gauss-Jordan method: So . By a recurrence we show that: : Initialisation : : : OK Hérédité : : OK 4. On a le system matrix following: . So Then: Exercise 3 is the general term of an alternating convergent Riemann series. Therefore is convergent. 2. here is the general term of a convergent Riemann series. So is convergent. 3. here is the general term of a convergent Riemann series. [...]
[...] So is convergent. and donc from where This is the general term of a divergent series. So the series is divergent. Exercice 4 1. with the general term of a convergent Riemann series. So is convergent. This is the general term of a convergent Riemann series. Therefore is convergent. 2. On decompose in simple elements: then , from where . Donc . So . 3. et . [...]
