I have been reading Concrete Mathematics by Donald Knuth. Upon reading chapter 2 on page 29, I came across the following sum : $H_n=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}=$$\sum_{k=1}^n \frac{1}{k}$.
From this I deduce that $H_n$ is defined for any integer $\geq 1$.
However, on page 56, we have the following: $\sum_{}^{} xH_x \delta x = \frac{x^{\underline2}}{2}H_x - \frac{x^{\underline2}}{4} + C$ and that $\sum_{0}^{n-1} kH_k = \sum_{0}^{n} xH_x \delta x =\frac{n^{\underline2}}{2}(H_n - \frac{1}{2})$.
That is, $\frac{x^{\underline2}}{2}H_x - \frac{x^{\underline2}}{4} + C \bigg|_{0}^{n} = \frac{n^{\underline2}}{2}(H_n - \frac{1}{2})$.
But this last line requires us to compute $(\frac{n^{\underline2}}{2}H_n - \frac{n^{\underline2}}{4} + C) - (\frac{0^{\underline2}}{2}H_0 - \frac{0^{\underline2}}{4} + C)$. But $H_0$ is not defined. So how can we correctly compute this last line?
Actually, there is a natural way to define $H_0$ according to the definition of $H_n$. A summation like $\sum_{k=1}^0 \frac{1}{k}$ makes sense: you are adding up $\frac{1}{k}$ for all values of $k$ such that $1\leq k\leq 0$. There are no such values of $k$, so you are adding up no numbers. When you add up no numbers, you get $0$, so $H_0=0$.