For $x\ge 0$, the geometric distribution tail givesFor $x<0$, $P(T_k\le x)=0$. The limiting cumulative distribution function is therefore $0$ for negative $x$ and $1-e^{-x}$ for nonnegative $x$, and it is continuous everywhere. This proveswith rate parameter $1$ and mean $1$.

At a fixed index, independence and uniformity giveThe match indicators are independent Bernoulli trials. Thus the waiting times between successive matches are independent geometric variables on $\{1,2,\dots \}$ with success probability $p$. Their moment-generating function isprovided $(1-p)e^t<1$. Since $S_m$ is the sum of $m$ such waiting times,For $k>1$, the finite domain is $t<\log (k/(k-1))$. For $k=1$, every index is a match, so $S_m=m$ deterministically and the moment-generating function is $e^{mt}$ for every real $t$.

Expand $x^x=e^{x\log x}$. The absolute values of all terms have total integralThe series converges, for instance because $j^{-j}\le 2^{-j}$ for $j\ge 2$. Part (b) applied to the absolute values, followed by subtraction of positive and negative parts, therefore justifies termwise integration. It gives the other Sophomore's dream identity:Equivalently, the partial sums are dominated by $e^{|x\log x|}\le e^{1/e}$, so the dominated convergence theorem gives the same justification.