The sum of infinite series ....... is

Engineering

Mathematics

Introduction to Determinants

Question

The sum of infinite series $∣ \begin{array}{cc} 1 & 2 \\ 6 & 4 \end{array} ∣ + ∣ \begin{array}{cc} \frac{1}{2} & 2 \\ 2 & 4 \end{array} ∣ + ∣ \begin{array}{cc} \frac{1}{4} & 2 \\ \frac{2}{3} & 4 \end{array} ∣ +$ ....... is

– 10

∞

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$∣ \begin{array}{cc} 1 & 2 \\ 6 & 4 \end{array} ∣ + ∣ \begin{matrix} \frac{1}{2} & 2 \\ 2 & 4 \end{matrix} ∣ + ∣ \begin{matrix} \frac{1}{4} & 2 \\ \frac{2}{3} & 4 \end{matrix} ∣ + .. .. .. .$
$= 4 - 12 + 2 - 4 + 1 - \frac{4}{3} + .. .. .$
$= (4 + 2 + 1 + .. .) - (12 + 4 + \frac{4}{3} + .. ..)$
$= \frac{4}{1 - \frac{1}{2}} - \frac{12}{1 - \frac{1}{3}}$
= 8 – 18 = – 10

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