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

Question

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

Answer 1

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