Hi! I am not 100 % sure what you mean in every example. E.g. in a) do you mean or ? Anyway, both of these are definitely divergent for , since their absolute values are both bigger than , which is unbound and divergent.

And in b) do you mean:

,

,

or something else? The first and the last definition is definitely divergent, because they are both bigger than , which is definitely divergent. The second definition converges, because it can be rewritten as:

For going to infinity all terms vanish except .

c) is also convergent regardless if you mean or .

d) is convergent because for positive n:

e) is definitely divergent because for positive n: .