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There is a plane with 100 seats and we have 100 passengers entering the plane one after the other. The first one cannot find his ticket, so chooses a random (uniformly) seat. All the other passengers ...

The answer is "no", the limit does not exist, because (now I'm just collecting the comments already made...) consider the series $(2p_n-1)_n$, where $p_1,p_2,\dots$ denote all primes. We have
$$\...

Rewriting the recursion yields
$$1 = \sum\limits_{1\leq i \leq k} \frac{i\ f(i)}{k+i-1}.$$
By assuming $f(k)=\frac{1}{k \ln 2} (1+o(1))$ --- where the little o-Notation $o(1)$ simply means that it is ...

Assume the interval $[0,1)$ is countable. Then we can write down all numbers like that:
$a_0 = 0.a_{0,0}a_{0,1}a_{0,2}\dots$
$a_1 = 0.a_{1,0}a_{1,1}a_{1,2}\dots$
$a_2 = 0.a_{2,0}a_{2,1}a_{2,2}\dots$
$...

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