Ratio Test

Ratio Test

• If ${\displaystyle \lim_{n\to\infty}}|\frac{a_{n+1}}{a_n}|<1$, then $\sum\limits^\infty_{n=1}a_n$ converges (infact, it converges absolutely)
• If ${\displaystyle \lim_{n\to\infty}}|\frac{a_{n+1}}{a_n}|>1$, then $\sum\limits^\infty_{n=1}a_n$ diverges
• If ${\displaystyle \lim_{n\to\infty}}|\frac{a_{n+1}}{a_n}|=1$, then the test fails and we don't know whether the series converges or not

Identifying Clues

• the terms look like a ratio of two functions (rational function)
• the terms contain factorials(!) or powers