Determine convergence of 2*abs(an) from 1 to inf

[isukatphysics69]

Homework Statement

Homework Equations

The Attempt at a Solution

confused here, so my book seems to be saying that 2*abs(an) converges which i thought was bogus so i went over to symbolab and symbolab is saying it diverges which i agree with. Why is my book saying this? Am i misenterpereting what the book is saying?

 

[Mark44]

Homework Statement

View attachment 224803 View attachment 224804

Homework Equations

The Attempt at a Solution

confused here, so my book seems to be saying that 2*abs(an) converges which i thought was bogus so i went over to symbolab and symbolab is saying it diverges which i agree with. Why is my book saying this? Am i misenterpereting what the book is saying?

Yes, you are misinterpreting what the book is saying.
The series in the textbook and the series in the screen shot are completely different. What the book says is that if $\sum_{n = 1}^\infty |a_n|$ converges, then $\sum_{n = 1}^\infty a_n$ converges. The sum in the screen shot, $\sum_{n = 1}^\infty 2|n|$, diverges, so is not a counterexample for what the book states.

 

[isukatphysics69]

Yes, you are misinterpreting what the book is saying.
The series in the textbook and the series in the screen shot are completely different. What the book says is that if $\sum_{n = 1}^\infty |a_n|$ converges, then $\sum_{n = 1}^\infty a_n$ converges. The sum in the screen shot, $\sum_{n = 1}^\infty 2|n|$, diverges, so is not a counterexample for what the book states.

Ohhh i see lol. Ok that makes sense now thank you