D'Alembert criterion

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series

where each term is a real or complex number and a_n is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.

Given the following geometric series:

The quotient

of any two adjacent terms is ${\displaystyle {\frac {1}{2}}}$. The sum of the first m terms is given by:

As m increases, this converges to 1, so the sum of the series is 1. On the other hand given this geometric series:

The quotient ${\displaystyle {\frac {a_{n+1}}{a_{n}}}}$ of any two adjacent terms is 2. The sum of the first m terms is given by

which increases without bound as m increases, so this series diverges. More generally, the sum of the first m terms of the geometric series is given by:

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