How to evaluate Catalan number and other sums?

Use the Sum command.

restart; 
expr:= (-1)^i/(2*i+1)^2; 
Sum(expr,i=0..infinity); 
evalf(%,50); 
          0.91596559417721901505460351493238411077414937428167

Notice, if I used the sum command instead of the Sum command I get this result:

sum(expr,i=0..infinity); 
          Catalan