Section Summary

In this section we have studied series of constants, i.e., series that might represent specific values of functions that can be represented by series. However, a series cannot represent a specific value unless it converges. In particular, we saw that a series can't converge unless its terms approach `0.`

Our sample series were generated by substituting specific values of `x` into the Taylor series for `text(ln) text[(] 1 + x text[)]` and for `text(arctan) x`. Interesting series arise from choosing values of `x` at the endpoints of intervals of convergence, because it is not clear a priori whether such series converge or diverge. In fact, we found that the alternating harmonic series and Leibniz series both converge, but the harmonic series diverges. In particular, the harmonic series is the prototypical example of a series whose terms approach `0` but that nevertheless diverges.

In the next section we will formalize into a test for convergence what we have learned here about alternating series. We also will study why Taylor series have an interval of convergence. The outcome of that study will be a convergence test we can use for series that are not necessarily alternating.