What is the ratio test?

The ratio test is a convergence test used in calculus to determine the convergence or divergence of an infinite series.

The ratio test states that for a series (\sum_{n=1}^{\infty} a_{n}), if the limit

[L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_{n}}\right|]

exists, then:

• If (0 \leq L < 1), the series converges absolutely.
• If (L > 1) or (L = \infty), the series diverges.
• If (L = 1) or the limit does not exist, the ratio test is inconclusive and other convergence tests should be used.

The ratio test is particularly useful for series with terms that involve factorials or exponentials, where the limit of the ratio simplifies to a more manageable form.

One important thing to note is that the ratio test does not necessarily determine convergence or divergence for all series, as there are certain series for which the ratio test may be inconclusive. In such cases, it is recommended to use other convergence tests to determine the convergence or divergence of the series.