Suppose I want to get a really good idea about the behaviour of gamma(z) near z=1. What about doing the following:
Say this is a relief of the Gamma function in the third Jacobian theta function. Gamma could be replaced with any function whose limit as z approaches 1 is 1. Places where the relief is not one are places where the numerator differs from the denominator, thus taking a very fine microscope to the way the gamma function behaves.
Similarly, we can do the same for infinite sums. All you need is a function which goes to zero as its argument goes to zero. Sometimes you can even do self-reliefs:
So the self-relief of sine in sine would be: