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FIBONACCI AND LUCAS SERIES

In this chapter, we study the convergence of some interesting Fibonacci and Lucas series, and evaluate them when convergent.

24.1 A FIBONACCI SERIES

To begin with, suppose we place successively every Fibonacci number c024-math-001 after a decimal point, so its units digit falls in the c024-math-002st decimal place. The resulting real number, to our great surprise, is the decimal expansion of the rational number c024-math-003. Be sure to account for the carries:

equation

that is, c024-math-004. F. Stancliff discovered this result in 1953 [537].

To establish this fact, we need to study the convergence of the Fibonacci series

where k is a positive integer.

Suppose the series converges. Then, by Binet's formula,

Notice that the denominator of the RHS of equation (24.3) is the characteristic polynomial of ...

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