Question 25
Let $(a_n)_{n \ge 1}$ be a sequence of integers larger than $1$ and
$$
b_n = 1 - \frac{1}{a_1} + \frac{1}{a_1 a_2} - \dots + (-1)^n \frac{1}{a_1 a_2 \dots a_n}, \quad n = 1, 2, 3, \dots
$$
Prove that:
a) the sequence $(b_n)_{n \ge 1}$ is convergent;
b) if $(a_n)_{n \ge 1}$ is unbounded, then the limit of the sequence $(b_n)_{n \ge 1}$ is irrational;
c) there exists a bounded sequence $(a_n)_{n \ge 1}$ so that the limit of the sequence $(b_n)_{n \ge 1}$ is irrational.
$$
b_n = 1 - \frac{1}{a_1} + \frac{1}{a_1 a_2} - \dots + (-1)^n \frac{1}{a_1 a_2 \dots a_n}, \quad n = 1, 2, 3, \dots
$$
Prove that:
a) the sequence $(b_n)_{n \ge 1}$ is convergent;
b) if $(a_n)_{n \ge 1}$ is unbounded, then the limit of the sequence $(b_n)_{n \ge 1}$ is irrational;
c) there exists a bounded sequence $(a_n)_{n \ge 1}$ so that the limit of the sequence $(b_n)_{n \ge 1}$ is irrational.