TAOCP 1.2.3 Exercise 11

Section 1.2.3: Sums and Products

Exercise 11. [03] What should the right-hand side of Eq. (14) be if $x = 1$?

Verified: yes
Solve time: 34s

If $x = 1$, the left-hand side of Eq. (14) becomes

$\sum_{0 \le j \le n} a = a + a + \cdots + a,$

with exactly $n+1$ terms. Therefore

$\sum_{0 \le j \le n} a = a(n+1).$

Hence the right-hand side of Eq. (14) should be replaced by

$\boxed{a(n+1)}.$