Sigma with Or Underneath

Question

This is the notation found in Spivak's Calculus: Chapter 23: Infinite Series Theorem 9

What does $\displaystyle \sum_{i \text{ or }j > L} |a_i||b_j| $ mean?

Answer 1

Whatever is underneath a summation sign tells you what you sum over. Formally, every such notation basically means this:

Let $A$ be a set of indices. A summation of some objects over the indexing set $A$ would be denoted (prefixed, really) by

$$\sum_{\alpha \in A}$$

In this sense, we can understand the notation $$\sum_{i=1}^n$$ to mean $$\sum_{i \in \{1,\ldots,n\}}.$$

Similarly in your case, $$\sum_{i~\text{or}~j > L}$$ indicates that your sum is the indexing set $A = \{(i,j): i > L~\text{or}~ j > L\}$. When you are performing the summation, you should think about all possible ordered pairs $(i,j)$ and include in the sum only those which satisfy $i > L$ or $j > L$. (Or both, of course.)