Short answers, pulled from the story.
A series in mathematics is, roughly speaking, the addition of infinitely many terms, one after another. The study of series is a major part of calculus and its generalization, mathematical analysis, and series appear across physics, computer science, statistics, and finance.
A series converges, or is summable, when the sequence of its partial sums approaches a fixed limit, and that limit is called the sum of the series. When the partial sums have no limit, the series diverges. A series with only finitely many nonzero terms always converges.
A series is absolutely convergent when the series of the absolute values of its terms also converges, and such a series keeps the same sum under any rearrangement. A conditionally convergent series converges but not absolutely, and by the Riemann series theorem its terms can be rearranged to reach any real number or to diverge. The alternating harmonic series, which sums to the natural logarithm of 2, is conditionally convergent.
Zeno's paradoxes, most famously Achilles and the tortoise, treated continuous motion as infinitely many sub-races each taking finite time, which seemed to make catching the tortoise impossible. The mathematical side of the paradox is resolved because that series has infinitely many terms but a finite sum, equal to the time Achilles needs to catch up. This resolution came through the concept of a limit in the 17th century, especially in Isaac Newton's early calculus.
Archimedes produced the first known summation of an infinite series using the method of exhaustion, and Nicole Oresme proved the divergence of the harmonic series in the 14th century. The Kerala school in medieval India studied series around 1350 CE, James Gregory and Brook Taylor advanced them in the 17th and early 18th centuries, and rigor came in the 19th century through Carl Friedrich Gauss and Augustin-Louis Cauchy.
Common convergence tests include the nth-term test, the direct and limit comparison tests, the ratio test, the root test, the integral test, and Cauchy's condensation test for series with non-negative terms. For conditional convergence, the alternating series test, Dirichlet's test, and Abel's test are used.