consider the series ∑∞n = 1an. ∑n = 1∞an. let ρ = limn→∞n|an|. according to the root test, if ρ > 1 or ρ =…

consider the series ∑∞n = 1an. ∑n = 1∞an. let ρ = limn→∞n|an|. according to the root test, if ρ > 1 or ρ = ∞, then ∑n = 1∞an converges absolutely. ∑n = 1∞an converges conditionally. ∑n = 1∞an diverges. the test does not provide any information.

consider the series ∑∞n = 1an. ∑n = 1∞an. let ρ = limn→∞n|an|. according to the root test, if ρ > 1 or ρ = ∞, 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 rules

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 $\rho>1$ or $\rho = \infty$, the terms $\vert a_n\vert$ do not approach 0 as $n\rightarrow\infty$.

Step2: Apply divergence criterion

A necessary condition for the convergence of a series $\sum_{n = 1}^{\infty}a_n$ is that $\lim_{n\rightarrow\infty}a_n = 0$. When $\rho>1$ or $\rho=\infty$, $\lim_{n\rightarrow\infty}\vert a_n\vert\neq0$, so $\lim_{n\rightarrow\infty}a_n\neq0$. Thus, the series $\sum_{n = 1}^{\infty}a_n$ diverges.

Answer:

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