TAOCP 1.2.3 Exercise 1
By definition, a sum of the form $a_1 + a_2 + \cdots + a_0$ is empty because the upper limit is less than the lower limit.
Section 1.2.3: Sums and Products
Exercise 1. ▶ [10] The text says that $a_1 + a_2 + \cdots + a_0 = 0$. What, then, is $a_2 + \cdots + a_0$?
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By definition, a sum of the form $a_1 + a_2 + \cdots + a_0$ is empty because the upper limit is less than the lower limit. According to the conventions in Section 1.2.3, the sum over an empty range is $0$.
Similarly, the sum $a_2 + \cdots + a_0$ is also over an empty range, since $2 > 0$. Therefore, by the same convention, we have
$a_2 + \cdots + a_0 = 0.$
This completes the proof.
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