Let's start fron the Newton expansion...

(1)

... that can be rearranged as...

(2)

... where...

(3)

It is easy to verify from (3) that is...

(4)

From the last of (4) and (2) we derive the well known result...

(5)

Now we will use (2) for computing the difference between two consecutive terms of the sequence ...

(6)

Since the (6) is a series with alternate decreasing terms and the first is positive, the sequence is increasing and, taking into account (5), we conclude that...

(7)

... so that half of your first question is answered...

Kind regards