For a positive real number p, the tower of exponents p^pp a continues indefinitely and the expression is ambiguous. The tower could be built from the top as the limit of the sequence {p^p, (p^p)^p, ((p^p)^p)^p, . . .}, in which case the sequence is defined recursively as

a_{n + 1} = a^p_n (building from the top,                                             (1)

where a₁ = p^p. The tower could also be built from the bottom as the limit of the sequence {p^p, p^(pp), p^(p(pp)), . . .} in which case the sequence is defined recursively as

a_{n + 1} = p^an (building from the bottom),                                   (2)

where again a₁ = p.

a. Estimate the value of the tower with p = 0.5 by building from the top. That is, use tables to estimate the limit of the sequence defined recursively by (1) with p = 0.5. Estimate the maximum value of p > 0 for which the sequence has a limit.

b. Estimate the value of the tower with p = 1.2 by building from the bottom. That is, use tables to estimate the limit of the sequence defined recursively by (2) with p = 1.2. Estimate the maximum value of p > 1 for which the sequence has a limit.