Series Divergence via Asymptotic Behavior

An example with the harmonic series

Students have been asking many questions about series recently. One that came up was about relating a sequence’s asymptotic behavior to the summability of a series. Two key steps are needed. First, ask “what does the sequence look approximately like when n is big?” In this case, {s_n} looks like L/n. With this intuition, we then choose ε in the sequence convergence definition for which we get a good approximation of this behavior when n is big (e.g. s_n > L / 2n).

Check out a full write-up in these slides.