Adding halves forever: how geometric series tame infinity
Add a half, then a quarter, then an eighth, and keep going without end. Despite infinitely many terms, the total settles on a definite value. That is a geometric series, where each term is the previous one multiplied by the same ratio, and whether it settles or explodes depends entirely on that ratio.
Build a sequence by starting with some number and repeatedly multiplying by a fixed constant, the common ratio, and you have a geometric progression; summing it gives a geometric series. Each term sits at the geometric mean of its two neighbours, mirroring how arithmetic series work with ordinary averages. When the ratio exceeds one it is often called a growth rate, and below one a decay rate, and fields borrow their own words: economists talk about inflation, deflation, interest and rates of return.
With infinitely many terms, the series may converge to a value or diverge. The magnitude of the ratio decides. If it is less than one, the terms shrink toward zero and the partial sums close in on a limit. If it is greater, the terms balloon and the sum runs away. Right at the boundary, strange things happen: Grandi's series flips back and forth between two values forever, and a ratio equal to the imaginary unit makes the partial sums circle endlessly around a few complex numbers. Even a converging series slows down as the ratio approaches one in size, since the error shrinks by a factor equal to the ratio at every step.
A handy trick follows. A repeating decimal such as 0.7777 recurring is really seven tenths plus seven hundredths and so on, a geometric series, which lets it be rewritten as a ratio of two whole numbers.
Zeno's 5th-century BCE paradoxes of motion have been read as puzzles about such series, but Greek mathematicians studied them formally a century or two later; Archimedes used one in the 3rd century BCE to find the area inside a parabola. Today they turn up in mathematical finance, in measuring the areas of fractals and across computer science, and mathematicians extend them from ordinary numbers to matrices, functions and abstract algebraic structures.
Source: Geometric series
