Ratio Test

The ratio test is useful to conclude the divergence/convergence of a Series containing either exponents or factorials.

Theorem §

Ratio Test

For some sequence ($(a_n{)}_{n\in \mathbb{N}}$), let:

$$L=\underset{n\to \infty }{\mathrm{limsup}}(\left|\frac{a_{n+1}}{a_n}\right|)$$

Then:

$$\{\begin{array}{ll}L<1,& {\sum }_{n=1}^{\infty }(a_n{)}_{}\text{ is absolutely convergent (and hence convergent)}\\ \exists N,\forall n>N\wedge L\ge 1,& {\sum }_{n=1}^{\infty }(a_n{)}_{}\text{ is divergent}\end{array}$$

where ${\mathrm{limsup}}_{n\to \infty }$ is the limit superior.

Proof §

#todo for convergence, compared to geometric series

For divergence, do divergence test