Comp.145 Example on Sequences, Limits and Series The sequences {x_{n}} and {y_{n}} are defined for n = 0, 1 … and a given function f(x) and constant a by

x_{n+1} = f(x_{n}) ,  y_{n} = ∑_{k=0}^{n} x_{k}  with x_{0} = a .

If the sequence {x_{n}} tends to a limit X, find an equation satisfied by X. Consider the behaviour of {x_{n}} and {y_{n}} as n → ∞ for the following cases:

(i) f(x) = bx, for some constant b, (ii) f(x) = x/(1 + x) , [Hint Relate 1/x_{n+1} to 1/x_{n}.] (iii) f(x) = 2x − x^{2} . [Hint: Relate x_{n+1} − 1 to x_{n} − 1.] In each case, find an explicit formula for x_{n}, and state carefully the values of a and b for which the sequences x_{n} and y_{n} converge.