Question

Read the section and make your own summary of material.

Short answer

The Ratio Test states that, if $\sum _{k=1}^{\infty }{a}_k$is a series with positive terms and $\rho =\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}$exists then,

(a) the series converges when$\rho <1.$

(b) the series diverges when$\rho >1.$

(c) the test is inconclusive when$\rho =1.$

The Root Test states that, if $\sum _{k=1}^{\infty }{a}_k$is a series with positive terms and $\rho =\underset{k\to \infty }{\lim }\sqrt[k]{a_k}=\underset{k\to \infty }{\lim }{\left(a_k\right)}^{\frac{1}{k}}$exists then,

(a) the series converges when$\rho <1.$

(b) the series diverges when$\rho >1.$

(c) the test is inconclusive when$\rho =1.$

Step-by-step solution

Step 1. Given information.

The given topic of the section is The Ratio and Root Tests.

Step 2. Summary of the section.

The Ratio Test states that if $\sum _{k=1}^{\infty }{a}_k$is a series with positive terms and $\rho =\underset{k\to \infty }{\lim }\frac{a_{k+1}}{a_k}$exists then,

(a) the series converges when$\rho <1.$

(b) the series diverges when$\rho >1$

(c) the test is inconclusive when$\rho =1.$

The Root Test states that if $\sum _{k=1}^{\infty }{a}_k$is a series with positive terms and $\rho =\underset{k\to \infty }{\lim }\sqrt[k]{a_k}=\underset{k\to \infty }{\lim }{\left(a_k\right)}^{\frac{1}{k}}$exists then,

(a) the series converges when$\rho <1.$

(b) the series diverges when$\rho >1.$

(c) the test is inconclusive when$\rho =1.$