Sequences and Series

Abstract

Sequences and series have fascinated people for thousands of years. They are arrows pointing at the unreachable infinite. Aristotle described the paradoxes due to Zeno, of Achilles racing the tortoise and of “dichotomy,” both of which are answerable today as questions about infinite series. And Archimedes understood that the geometric series $1 + {1 \over 4} + {1 \over {{4^2}}} + {1 \over {{4^3}}} + ...$ was the number 4/3. But there was very little more than that known, in theory or practice, to guide Isaac Newton when he went to work on the calculus. He used series in wholely new ways, applying his techniques of integration and differentiation to them term by term.