I'm currently reading the book Concrete Mathematics, and I feel like I cannot understand the conditions author put on 5.12. The text is:
$$(x + y)^r = \sum_k {r \choose k} x ^ ky^{r-k} $$$$\text{integer r} \ge \text{0 or }|x/y|\lt 1$$
The sum is over all integers k; but it is really a nite sum when r is a nonnegative integer, because all terms are zero except those with $0 \le k \le r$. On the other hand, the theorem is also valid when r is negative, or even when r is an arbitrary real or complex number. In such cases the sum really is infinite, and we must have $|x/y|\lt 1$ to guarantee the sum's absolute convergence. Two special cases of the binomial theorem are worth special attention, even though they are extremely simple. If x = y = 1 and r = n is nonnegative, we get $$2^n = {n \choose 0} + {n \choose 1} + \cdots+{n \choose n}, \text{integer } n \ge 0$$
I am confused with the $|x/y|\lt 1$ condition. I think it means "the absolute value of x/y", but how can $x=y=1$ be a case of the formula if $|x/y|\lt 1$? If id doesn't mean absolute value, what does it mean?
It's my first time using MathJax. Sorry for the bad formatting.
Actually I made an careless mistake by ignoring the "or" between the two conditions. Since the first condition is satisfied, the second one don't need to be satisfied. Thank all of you for helping me on it.