Does this series always diverges?

Question

Let a_n be a non identically zero arithmetic progression. Then, is it true that Sum (1/a_n) diverges?

Answer 1

a_n = a_1 + r * (n-1)

= (n-1) * (r+a_1/(n-1))

If n is large, | r+a_1/(n-1) | < 2r.

So a_n < (n-1) * 2r for n large.

So 1/a_n > 1/(2r) * (1/(n-1))

So, there is a C = 1/(2r) such that 1/a_n > C/(n-1) for large n.

Since sum (1/(n-1)) diverges, sum(1/a_n) diverges.

Answer 2

an= ao+n*r ( positive terms)
bn= 1/an is of the same class as 1/n which diverges