Prove that$\sum_{n=1}^{\infty} 2^{-2^{n}}$ converges to an irrational limit.

Question

Consider the development of the limit in base $2$. Is it periodic? — Jean-Claude Arbaut, Nov 23 '18 at 22:39
showing it is even transcendental should not be too hard, given that it has so good approximations — Hagen von Eitzen, Nov 23 '18 at 22:57

score 4 · Accepted Answer · answered Nov 23 '18 at 22:45

4

The binary representation $0.01010001\cdots$ doesn't terminate or repeat.

answered Nov 23 '18 at 22:45

J.G.

1 Answers1