Let’s say you have a complex expression, say with a few gnarly infinite sums and want to know its value. Typically, your first step typically is to show that i

Let’s say you have a complex expression, say with a few gnarly infinite sums and want to know its value. Typically, your first step typically is to show that it is finite because that often is a lot easier, and if it turns out not being finite, you’ve spared yourself from doing some complex calculations. The typical approach to show that some expression is finite is to replace difficult parts of the expression by simpler ones that certainly aren’t smaller. If the changed, definitely not smaller, expression is finite, so must be the original one. As a simple example, a rouge estimate of sin(x) + cos(x) says both sin(x) and *cos(x) are in [-1,1], so their sum is in [-2,2]. That shows the expression is finite for all x. (Actually, the sum is in [-√2, √2], but if you don’t need to be that precise, why bother?) Similarly, I wouldn’t know what the sum of i=1 to infinity of |sin(i)|/2^i is, but each term is positive and not greater than 1/2^i, so the sum isn’t greater than 1, and that series must converge.