Chapter 1.6, Problem 30P

Prove the ratio test. Hint: If $|a_{n+1}/a_n|\to <1$, take $\sigma$ that $p<\sigma <1$.

Then $|a_{n+1}/a_n|<\sigma$ if n is large, say $n\ge N$.

This means that we have $|a|<\sigma |a_N|,|a_{N+2}|<\sigma |a_{N+1}|<{\sigma }^2|a_N|,$ and so on. Compare with the geometric series ${\displaystyle \underset{n=1}{\overset{\infty }{\sum }}{\sigma }^n\left|a_N\right|.}$

Also prove that a series with p > 1 diverges. Hint: Take $p>\sigma >1$, and use the preliminary test.