Error in book ?

An error in a textbook? I've never seen such a thing before!!!

Why do you think a book is constantly revised with newer editions?

 

How did you evaluate an infinite sum in a finite time?

 

If the result is true when [tex]|z| < 1/\sqrt 2[/tex], then it is true when [tex]|z| < 1/2[/tex].

So what's the error?

 

So you put a few terms into a calculator and decided that was the behaviour for infinitely many terms? That isn't how you prove things.

It does look like a mistake, by the way.

 

Yea matt I do that sometimes as well, it doesn't prove anything at all, but sometimes its obvious what its converging to, can give you a tiny prod in the right direction.

 

If you check it, then you demonstrate it to be true, i.e. you prove it. Numerical things like this merely suggest, they do not verify.

 

We're trying to be helpful, please note these are people who are not getting paid to answer your questions! These people are very knowledgeable people who could better use their time! If you think they are being pedantic, hack it, your getting free advice from smart people!

 

you have been answered on this topic.

it's irony. mistakes in textbooks are very frequent, and almost not worth pointing out.

this was pointing out that it wasn't clear if you were asking for a sufficient condition or a necessary and sufficient condition.

If you had said that the book asserted the radius of convergence was something, then that would be different. What you wrote is, if as litereally appears in the book, perfectly correct. That sum does converge for |z|<1/2. That is not the radius of convergence, but that is strictly a different matter from what you asked.