For the following task I only get the numerical solution 2.356... :
sum(n=1 to oo)[arctan(2/n^2)]
The known "closed" result is 3*pi/4.

For the following limit determination, I do not get the known result 2, but only a term with LerchPhi. Please give me some advice.
lim(n-->oo)[(n/(2^n))*sum(k=1 to n) (2^k/k)]

I am very interested in problems of integration and limit determination. In order to be able to continue other work with the help of Maple, I would like some advice on solving the following problem. The solution I know of using pen and paper is very tiring, perhaps Maple can make it easier:

To calculate Int(from 0 to 1)[ln(x)*ln(1-x)]dx
or, for obvious reasons, formulated differently:
Let eps, delta>0. Calculate
lim(eps, delta-->0)Int(0+eps to 1-delta)[ln(x)*ln(1-x)]dx

One of the most difficult/nice tasks from my collection of old problems:
Calculate Int(from 0 to 1)Int(from 0 to 1)[1/(1-x*y)]dxdy .

Let eps be a real number greater than zero. Calculate the limit lim(x-->oo)[x^(1-eps)*Integral(from x to x+1) sin(t^2)dt] .