A divergent sequence where the reciprocal diverges as well

Question

A divergent sequence (bn) with b_n≠0 for all n such that (1/b_n) diverges.

Or state that it is impossible.

Answer 1

A divergent sequence is a sequence that does not converge.

Look at the following sequence:

b₀ = 1

b_n = n , if n is even and n > 0,

b_n = 1/n , if n is odd.

The first terms of this sequence are 1, 1, 2, 1/3, 4, 1/5, 6, 1/7. This sequence clearly diverges since the subsequence of even indices tends to infinity. Furthermore, let a_n = 1/b_n. This sequence also diverges since now the subsequence of odd indices tends to infinity.