MathDB

For any positive integer

k

consider the sequence

a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},

where there are

n

square-root signs on the right-hand side.
(a) Show that the sequence converges, for every fixed integer

k\ge 1

. (b) Find

k

such that the limit is an integer. Furthermore, prove that if

k

is odd, then the limit is irrational.

Problem Statement