Ratio test for convergence of a series

[Bashyboy]

Homework Statement

I attached a file that includes the author's solution, and some of my work.

Homework Equations

The Attempt at a Solution

 

[Mark44]

Homework Statement

I attached a file that includes the author's solution, and some of my work.

Homework Equations

The Attempt at a Solution

The denominator in the series is not (2n + 1)! It is the product of the odd integers from 1 to 2n + 1.

 

[Bashyboy]

But can't you write that as (2n + 1)! ?

 

[Mark44]

No, not at all.
(2n + 1)! = 1 * 2 * 3 * ... * (2n - 1) * (2n) * (2n + 1)

Your expression skips all the even integers.

A simple example is 5! vs. 1 * 3 * 5. Clearly they aren't equal.