What does it mean to majorize a quotient?

Question

This is from a proof regarding the uniform and absolute convergence of a power series, from Krantz's Real Analysis and Foundations:

We may take the compact subset of $\mathcal{I}$ to be $K = [c-s,c+s]$ for some number $0 < s < r$. For $x \in K$ it then holds that $$\sum_{j=0}^\infty |a_j(x-c)^j| = \sum_{j=0}^\infty |a_j(d-c)^j|\left|\frac{x-c}{d-c}\right|^j$$

In the sum on the right, the first expression in absolute values is bound by some constant $C$ (by the convergence hypothesis). The quotient in absolute values is majorized by $L = s/r < 1$. The series on the right is thus dominated by $$\sum_{j=0}^\infty C \cdot L^j$$

I have no idea what "majorized" means here. If someone were to ask me to majorize that quotient, I would probably let $d$ get arbitrarily close to $c$.

In case it's relevant, $\sum_{j=0}^\infty a_j(x-c)^j$ converges when $x = d \neq c$, and $r = |d-c|$.

What does it mean to majorize a quotient (majorize a series??)

Answer 1

Note: The term majorize is a synonym for bounded above.

So, when we read that a quotient $Q=\left|\frac{x-c}{d-c}\right|^j$ is majorized by some value $L$, it simply means that $$Q\leq L$$

The same holds when a series is majorized by some value $L$. It simply means that the value which is represented by the series is bounded above by $L$.