Please refer to this link. It is Erdos and Renyi's first paper on Random Graphs (1959). I am trying to work through it.

I'm struggling with equations (16), (17) and (21).

(16)

• I'm not sure why they are using those two bounds in the summation? It seems redundant
• Why they did not use the result from (15) in (16) in the summation. They only used (14)?

(17)

• Why does $\log\log(n)$ replace $M$ in (17)?

(21)

• What does the $(-1)^k$ figure do? Why is it necessary?