Review of Complex Analysis

We’ve rearranged the terms from a series for log 2 and obtained half the original value. Until the early 19th century, the evaluation of infinite series was not troubled by rearrangement issues as above, since there wasn’t a clear distinction between two issues: defining convergence of a series and computing a series. An infinite series needs a precise defining algorithm, such as taking a limit of partial sums via an enumeration of the addends, upon which other summation methods may or may not be comparable. This aspect of infinite series is at least historically tied up with zeta and L-functions, because it was resolved by Dirichlet in his (first) paper on infinitude of primes in arithmetic progressions, where he introduced the notion of absolutely convergent series to tame the rearrangement problem. Recall that a series ∑ zn of complex numbers is called absolutely convergent if the series ∑ |zn| of absolute values of the terms converges. This condition