# 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:**

- Both the columns and rows extend to infinity.
- In each row, the numerators are the terms of the Fibonacci sequence, while the denominators are the successive powers of the row number.
- 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.