Ratio Tests and Their Refinements for Infinite Series

All flashcards

Ordinary ratio-test limit

$L=\lim_{n\to\infty}|a_{n+1}/a_n|$, when the limit exists.

Ratio-test boundary case

When $L=1$, the ordinary ratio test is inconclusive.

Refined ratio test for convergence

For $q_n=|a_{n+1}/a_n|$, $\limsup q_n<1$ proves absolute convergence.

Raabe parameter

$\rho_n=n(a_n/a_{n+1}-1)$; a limit above 1 proves convergence, and one below 1 proves divergence.

Kummer convergence criterion

For positive $a_n$ and $\zeta_n$, a positive eventual lower bound on $\zeta_n(a_n/a_{n+1})-\zeta_{n+1}$ proves convergence.

Quiz questions

1. Let $q_n=|a_{n+1}/a_n|$ and $R=\limsup_{n\to\infty}q_n$. What follows if $R<1$?Answer: The series converges absolutely.
2. In Kummer's test, let $a_n>0$, $\zeta_n>0$, and $\rho_n=\zeta_n(a_n/a_{n+1})-\zeta_{n+1}$. What does an eventual bound $\rho_n\ge c>0$ establish?Answer: Convergence of $\sum a_n$.
3. In Kummer's test, which additional condition supports divergence when $\rho_n\le0$ eventually?Answer: $\sum 1/\zeta_n$ diverges.
4. Suppose $a_n>0$ and $a_n/a_{n+1}=1+\rho/n+C_n/n^s$, where $s>1$ and $C_n$ is bounded. What does Gauss's test conclude when $\rho=1$?Answer: The series diverges.
5. If the ordinary ratio limit $L$ exists, how are the refined ratio bounds $R=\limsup q_n$ and $r=\liminf q_n$ related to it?Answer: $R=r=L$.