Two convergent geometric sequences, \( \{ u_1, u_2, u_3, ....\} \) and \( \{ v_1, v_2, v_3, ....\} \), of positive terms (common ratio, \( 0 \lt r \lt 1 \) ) are determined by the following values:
\( u_1=6, u_3=4 \) and \( v_1=7,v_2=5 \).
The difference, \(D\), between their sums to infinity is \( \lvert \frac{6}{1-\sqrt{\frac{2}{3}}} - \frac{7}{1-\frac{5}{7}} \rvert =8.197 \) (to 4 sig.figs)
Place the integers 2 to 7 in the boxes below, using each integer once, to determine two convergent geometric sequences of positive terms for which the value of \(D\) is as large as possible.
What is this maximum value of \(D\) to 4 sig.figs?
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