
## Section3.4Absolute and Conditional Convergence

We have now seen examples of series that converge and of series that diverge. But we haven't really discussed how robust the convergence of series is — that is, can we tweak the coefficients in some way while leaving the convergence unchanged. A good example of this is the series

\begin{gather*} \sum_{n=1}^\infty \left(\frac{1}{3} \right)^n \end{gather*}

This is a simple geometric series and we know it converges. We have also seen, as examples 3.3.20 and 3.3.21 showed us, that we can multiply or divide the $n^{\rm th}$ term by $n$ and it will still converge. We can even multiply the $n^{\rm th}$ term by $(-1)^n$ (making it an alternating series), and it will still converge. Pretty robust.

On the other hand, we have explored the Harmonic series and its relatives quite a lot and we know it is much more delicate. While

\begin{gather*} \sum_{n=1}^\infty \frac{1}{n} \end{gather*}

diverges, we also know the following two series converge:

\begin{align*} \sum_{n=1}^\infty \frac{1}{n^{1.00000001}} && \sum_{n=1}^\infty (-1)^n \frac{1}{n}. \end{align*}

This suggests that the divergence of the Harmonic series is much more delicate. In this section, we discuss one way to characterise this sort of delicate convergence — especially in the presence of changes of sign.