Examples of computing expectations of sums 11/9

In class I went over the proof that the expected value of the negative binomial NB(r,p) is $\frac{r}{p}$, by writing it as the sum of r geometric (p) random variables.

I showed how one would compute the expected number of cereal boxes to buy if there were N objects to be collected:

$$E(X)=N(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots +\frac{1}{N}) =N(log(N) + 0.577 +\frac{1}{2N})$$

And the number of matches in the matching problem can be put as the sum of the indicator variables X_i where X_i=1 if the ith person gets his/her hat. As $E(X_i)=P(X_i=1)=\frac{1}{N}$, we have

$$E(Y_n)= N \times \frac{1}{N}$$

Try this link for the matching problem applet