Let denote the Euler’s Gamma function and denote the digamma function. Evaluate
The exercise can also be found at Aops.com .
Let denote the – th harmonic number and let . Prove that
Let denote the -th harmonic number. Evaluate the limit
and of course we have the series represantation of the constant, namely this:
We conclude that the desired limit is just .
Let be a differential equation such that
Find an explicit formula of .
Making use of the initial condition we get that . Thus . This also means that has no roots in the domain given. Hence the initial condition gives us and the function follows to be .
Examine if the series
is an integer. Let and let be the distance from to the nearest integer. That is
It is immediate that can be expanded periodically with period . Since is an integer we can get that
and the conclusion follows.