How to determine if a problem is solvable by analytic methods? I am frequently coming across problems which are not susceptible to analytic methods. 
This has often been frustrating as I have been unable to predetermine if my time would be wasted with such methods. 
Is there a theory behind which problems are solvable by analytic methods and which are not?
An example of such a problem I've come across was the following:
"Find a real solution to the equation $n=(1+1/n)^n$ "

According to Wolfy,
the result is
$n \approx 2.293...$.
For a start,
if a problem involves
the solution of
$x e^x = a$
for $x$ in terms of $a$,
this can not be exactly solved
(except by using the
Lambert W function,
which is sort of cheating).
In your case,
I would play around
and see if
I could get an
initial approximation.
$1+1/n
=n^{1/n}
=e^{\log n/n}
\approx 1+\log n/n + \log^2(n)/(2n^2)+\text{ higher order terms}
$,
so,
as a first approximation,
$\log n = 1$,
or
$n = e$.
For the next approximation,
$1
=\log n+\log^2 n/(2n)
$,
and already this is a pain.
From these initial explorations,
I would guess that
no analytic expression
could be found for the root.
