Ratio Test

Theorem

1. Let $\sum a_n$ where $\forall n\in \mathbb{N},a_n\ne 0$
2. Defining ${\lim }_{n\to \infty }|\frac{a_{n+1}}{a_n}|=L\in \mathbb{R}$
3. If $L<1$, then $\sum a_n$ Absolute Convergence
4. If $L>1$ then, $\sum a_n$ Diverges
5. If $L=1$ then, this test is inconclusive. Go use another Convergence Test

Often good when used with:

• Factorial
• Exponential Function

Example

Prove ${\sum }_{n=1}^{\infty }\frac{e^n}{(2n)!}$ Converges.

Proof

1. We can use R.T
2. Consider ${\lim }_{n\to \infty }|\frac{a_{n+1}}{a_n}|$
3. $={\lim }_{n\to \infty }|\frac{\frac{e^{n+1}}{(2n+2)!}}{\frac{e^n}{(2n)!}}|$
4. $={\lim }_{n\to \infty }|\frac{e}{(2n+1)(2n+2)}|=0$
5. Then, as $0<1$, this means $\sum a_n$ converges