question 4 (1 point) consider the series σ∞n=1an. ∑n=1∞an. let ρ = limn→∞n|an|. according to the root test…

question 4 (1 point) consider the series σ∞n=1an. ∑n=1∞an. let ρ = limn→∞n|an|. according to the root test, if 0 ≤ ρ < 1, then ∑n=1∞an converges absolutely. ∑n=1∞an converges conditionally. ∑n=1∞an diverges. the test does not provide any information.

question 4 (1 point) consider the series σ∞n=1an. ∑n=1∞an. let ρ = limn→∞n|an|. according to the root test, if 0 ≤ ρ < 1, then ∑n=1∞an converges absolutely. ∑n=1∞an converges conditionally. ∑n=1∞an diverges. the test does not provide any information.

Answer

Explanation:

Step1: Recall Root - Test rule

The Root - Test states that for the series $\sum_{n = 1}^{\infty}a_n$ with $\rho=\lim_{n\rightarrow\infty}\sqrt[n]{\vert a_n\vert}$, if $0\leq\rho < 1$, the series converges absolutely.

Step2: Match with options

We see that when $0\leq\rho < 1$, the series $\sum_{n = 1}^{\infty}a_n$ converges absolutely according to the Root - Test.

Answer:

$\sum_{n = 1}^{\infty}a_n$ converges absolutely.