I find divergent series kind of fun. Getting useful information out of them seems like getting something from nothing, and that appeals to my psyche.
For some series, we are interested in the terms, not their convergence. The individual terms may count something we want to count, and fact that the function defined by the series diverges is of little interest to us. We still want to count what its terms represent. Such things are called formal power series.
There are some power series that we can define even where they diverge, especially if we allow our variable to be complex. Thus the geometric series diverges when x > 1 but its formula, makes sense everywhere except at x = 1 and we can use it to attribute meaning, of a sort, to the series, everywhere but at x = 1.
We can even take a divergent series and deduce how its n-th partial sum grows as a function of n.
Thus, as we have already claimed, the n-th partial sum of the harmonic series, , behaves like ln(n). We can ask, what happens to the difference between that partial sum and ln(n+1)? This difference is called Euler's constant (usually written as gamma).
Exercise 15.3 Use a spreadsheet to evaluate Euler's constant to 9 decimal places. Hint: you will have to extrapolate to do this. To do this you can evaluate this difference at say n = 2, 4, 8, 16, 32, 64,... and assume that the difference between the partial sum and ln(x) + Euler's constant is a power series in . Taking twice each difference less the preceding difference will eliminate the first term in this series, taking of each resulting term less of its predecessor will eliminate the second term, and so on.