Answer to Question #308838 in Differential Equations for Ramyth

Question

Solve the following system of equations by using the concept of matrices and determinants. 𝟓𝒙 + 𝟕𝒚 + 𝟐 = 𝟎 𝟒𝒙 + 𝟔𝒚 + 𝟑 = 𝟎 (b) Find whether the following series are convergent or divergent √ 𝟏 𝟒 + √ 𝟐 𝟔 + √ 𝟑 𝟖 + ⋯

Answer 1

The first item:

$\begin{cases} 5x + 7y + 2 = 0 \\ 4x + 6y + 3 = 0 \end{cases}$

$x = \frac{\begin{vmatrix} -2 & 7\\ -3 & 6 \end{vmatrix}} {\begin{vmatrix} 5 & 7\\ 4 & 6 \end{vmatrix}} = \frac{-12 + 21}{30 - 28} = \frac{9}2 = 4.5$

$y = \frac{\begin{vmatrix} 5 & -2\\ 4 & -3 \end{vmatrix}} {\begin{vmatrix} 5 & 7\\ 4 & 6 \end{vmatrix}} = \frac{-15 + 8}{30 - 28} = \frac{-7}2 = -3.5$

The second item:

$\frac{\sqrt{1}}{4} + \frac{\sqrt{2}}{6} + \frac{\sqrt{3}}{8} + \dots = \sum_{k=1}^{\infty} \frac{\sqrt{k}}{2k + 2} = \sum_{k=1}^{\infty} \frac{1}{2\sqrt{k} + \frac{2}{\sqrt{k}}} \ge \sum_{k=1}^{\infty} \frac{1}{4\sqrt{k}}$

The last sum series is divergent, so the initial one is divergent too.