Subtopic: Sequences and series – infinite series, tests for convergence
Question Statement:
According to D'Alembert's ratio test for a series of positive terms $\sum a_n$, if $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = L$, the series:
Options:
(A) Converges if $L < 1$ and diverges if $L > 1$
(B) Converges if $L > 1$ and diverges if $L < 1$
(C) Converges for all $L \ge 1$
(D) Diverges only when $L = 1$
Answer Key:
A
Step-by-Step Solution & Explanation
D'Alembert's ratio test states that for $\sum a_n$ with $a_n > 0$ and $L = \lim_{n \to \infty} \frac{a_{n+1}}{a_n}$:
- If $L < 1$, the series converges absolutely.
- If $L > 1$, the series diverges.
- If $L = 1$, the test is inconclusive.