Using the attached problem, I want to calculate a term with a constraint that m is not equal to n. How do I insert general constraints into Maple?

test.mw

I'm doing something wrong again in the attached file. Please advise.

test.mw

For practice, I would like to calculate the left-hand limit according to the attached file. The computer does not finish. The result pi^2/6 is known from a calculation "on foot".

Download test.mwtest.mw

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)]