I want to give the proof for those who were interested.

First of all, a very indirect proof of this result can be found in [1, Lemma 2].

References

[1] B.G. Zhang and J.S. Yu, Existence of positive solutions for neutral differential equations,

Science in China Series A35(1992), no. 11, 1306--1313. View PDF

I will here give a shorter proof for a more generalized form.

Lemma 1. Let be an increasing divergent sequence and .

Then

if and only if

provided that

.____________________________(*)

Proof. We shall consider two different cases.

- Let

.

In this case, the claim is true since both the integral and the sum diverge.- Let

.

If we define

for ,

then is decreasing on .

We compute that

where we have changed the order of integration in the last step.

Then an equivalent claim reads as follows.

if and only if .

Define by for .

Adopting the convention that the empty sum is , we get

for all ,

which proves that

and

diverge or converge together since by the assumption (*), we learn that

and

diverge or converge together.

This completes the proof.________________________________

Corollary 1. If in addition to the assumptions ofLemma 1, we have

.___________________________(**)

Then

if and only if .

Remark. InCorollary 1, if we let and for , we get

the result in the first post. Trivially, in this case (*) and (**) hold.