assume $ (a_n) _1 ^ infty $ is a growing sequence of positive real numbers with limit $ + infty $. Yes $ p> 0 $, shows CA $$ sum_ {n = 1} ^ {+ infty} frac {a_ {n + 1} – a_n} {a_ {n + 1} a_n ^ p} $$ converges.

Easy to make the cases $ p geqslant $ 1:

$$

sum_ {n = 1} ^ {+ infty} frac {a_ {n + 1} – a_n} {a_ {n + 1} a_n ^ p} leqslant sum_ {n = 1} ^ {+ infty } frac {a_ {n + 1} – a_n} {a_ {n + 1} a_n} = frac 1 {a_1} <+ infty.

$$

What about the case $ 0 <p <1 $? I'm pretty sure it would not cause a lot of problems, but I just can not understand it at the moment. All tips are welcome. Thank you in advance.