For which values of t > 1 does the expression t^t^t . . . make sense?

Hint: fix t > 1 and define a0 = 1 and an+1 = t^an for n 0. The question asks when this sequence has a limit.

(a) Show that an+1 > an for all n 1. What does this tell you?

(b) When L = lim an exists, solve for t in terms of L. Use this to find the optimal upper bound for those values of t for which the limit exists. What happens for larger t?

(c) For these values of t, show by induction that an is bounded above by e for all n 1. What does this tell you?