[vzqpqtrb] some slowly diverging and slowly converging integrals

evaluate these improper integrals from (some suitable lower bound avoiding undefined values) to infinity.

In[]:= Integrate[1/(x^(1)), x]

Out[]= Log[x]

In[]:= Integrate[1/(x^(1+1/n)), x]

Out[]= -(n/x^n^(-1))

the first integral diverges as x tends to infinity; the second converges.

NB: exponentiation is right associative, so the above expression is equal to -(n/x^(n^(-1))) = -n/(x^(1/n)) .

digression: set n=1 to show Riemann zeta(2) converges; Basel problem asks for its exact value.

In[]:= Integrate[1/(x*(Log[x])^(1)), x]

Out[]= Log[Log[x]]

In[]:= Integrate[1/(x*(Log[x])^(1+1/n)), x]

Out[]= -(n/Log[x]^n^(-1))

In[]:= Integrate[1/(x*Log[x]*(Log[Log[x]])^(1)), x]

Out[]= Log[Log[Log[x]]]

In[]:= Integrate[1/(x*Log[x]*(Log[Log[x]])^(1+1/n)), x]

Out[]= -(n/Log[Log[x]]^n^(-1))

it is nice that these all have simple form.

future post: infinite product of iterated log.

previously vaguely similar: twin primes and LogIntegral