TAOCP 1.2.3 Exercise 9

Exercise 9. ▶ [05] Is the derivation of Eq. (14) valid even if $n = -1$?

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Yes. The derivation remains valid when $n=-1$.

For $n=-1$, the sum in Eq. (14) is

$$ \sum_{0\le j\le -1} ax^j, $$

which is an empty sum and therefore has value $0$. Examine the derivation step by step. The relation

$$ \sum_{0\le j\le -1} ax^j = a+\sum_{1\le j\le -1} ax^j $$

is valid by rule (d), since the second sum is empty and hence equal to $0$. The subsequent transformations involve only empty sums, so the final equation obtained in the derivation is

$$ (1-x)\sum_{0\le j\le -1} ax^j = a-ax^0 = 0. $$

Since $x\ne1$, division by $1-x$ yields

$$ \sum_{0\le j\le -1} ax^j=0. $$

Equation (14) gives

$$ a\left(\frac{1-x^{(-1)+1}}{1-x}\right) = a\left(\frac{1-1}{1-x}\right) = 0, $$

which agrees with the value of the empty sum. Therefore the derivation and the resulting formula are valid for $n=-1$ as well.

$$ \boxed{\sum_{0\le j\le -1} ax^j=0} $$