=Dan Asimov The answer turns out to be exactly 1 - 1/2! + 1/3! - 1/4! + ... - 1/52! where k! denotes k(k-1)(k-1)...3.2.1.
I think you left off an initial 1/0! - ...
According to my calculation, (which may be wrong in the last decimal place) this comes out to .3678794411714424 to 16 decimal places (agreeing with 1/e at least through the 15th decimal place) though with more work this probability can be expressed precisely as a fraction.
Or a computer algebrar (here Macsyma): 193937806586896328746924473226467394949530139620339174273819171217 ------------------------------------------------------------------ 527177615496365219422618541545122659969212453861982208000000000000 = 0.3678794411714423215955237701614608674458... (after including that missing 0th term).