The Fibonacci Array

Calculus Level 5

\[\begin {array}{ccccccccccc} & \color{Red}{\frac{1}{1}} & + & \color{Red}{\frac{1}{1}} & + & \color{Red}{\frac{2}{1}} & + & \color{Red}{\frac{3}{1}} & + & \color{Red}{\frac{5}{1}} &+& ... \\ + & \color{Red}{\frac{1}{2}} & + & \frac{1}{4} & + & \frac{2}{8} & + & \frac{3}{16} & + & \frac{5}{32} &+& ... \\ + & \color{Red}{\frac{1}{3}} & + & \frac{1}{9} & + & \frac{2}{27} & + & \frac{3}{81} & + & \frac{5}{243} & +&... \\ + & \color{Red}{\frac{1}{4}} & + & \frac{1}{16} & + & \frac{2}{64} & + & \frac{3}{256} & + & \frac{5}{1024} &+& ... \\ + & \color{Red}{\frac{1}{5}} & + & \frac{1}{25} & + & \frac{2}{125} & + & \frac{3}{625} & + & \frac{5}{3125} & +& ... \\ + & \color{Red}{...} &+& ... &+& ... &+& ... &+& ... & +&...\\ {+} & \color{Red}{...} &+& ... &+& ... &+& ... &+& ... & +&... \end {array} \]

The sum above as shown diverges. However, when the column and row in red is removed, the sum converges to \(A\). Find \(\lfloor 10^{5} A \rfloor\).

Details:

  1. Both the columns and rows extend to infinity.
  2. In each row, the numerators are the terms of the Fibonacci sequence, while the denominators are the successive powers of the row number.
  3. In each column, the numerators are the same, and the denominators are terms of the form \(k^n\), \(n\) being kept constant every column, and \(k\) is a positive integer corresponding to the row number of the term.
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