There really isn't an answer. The Hp50 cannot evaluate that series to e, symbolically. Series evaluation is probably based on look-up tables, and there is only so much stuff like that that can be packed into a calculator (now).
The TI89 can evaluate that series to e, but it cannot evaluate the taylor series for e^x to e^x. Nor can the TI89 evaluate the sum of the even terms to cosh(1).
It can take a bit of creativity and knowledge to coax these CAS calculators to give the answer you want, particularly when the problems are difficult, but it is still much easier to do that than to solve problems purely with pen and paper.
They do have some series stored, though. For example, both the HP50 and TI89 can sum reciprocal even powers in terms of pi.
The second series there converges to x/(x-1)^2, when it does converge. The HP50g actually can sum that one symbolically, when x has a fixed value. But in general, there is a remainder term of the form x^n. The calculator then tries to find that limit (if it isn't zero, it can't converge), but since x is undetermined it can't find the limit. (It depends on whether x's absolute value is more or less than 1)
Note that the lim function takes two arguments: The expression, and the value x tends to. It does not, in other words, take as a third argument the variable the limit applies to; it assumes the variable is the independent variable. So I presume what the calculator is doing behind the scenes is temporarily turning the independent variable to 'n' to take the limit. Maybe it is supposed to turn the independent variable back to 'x' when it's done, but the error prevents that. In any case, if you sum over the variable x, this won't happen.
