Why can't I send Emails from iPad apps?

When I am using iBooks, or Evernote, or Hootsuite, or just about any application on my iPad that allows me to Email items from within the application, the following happens when I try to send an Email:

1) I click the share button.

2) Choose Emaill

3) Write the Email.

4) Click Send

5) Hear swoosh sound and it appears the Email is sent.

However, the Email is never actually sent. I've tested this mutiple times to various Email addresses and it just isn't sending the Email. The only exception is when I send Emails from within the Gmail app, in which case they send fine.

After discovering this I took our my iPhone 5 and tried to do the same from there with the same applications. I was able to send Emails from within the iPhone applications just fine so this problem is specific to my iPad.

I checked my mail settings and didn't see anything out of order. As I said, I can successfully send Email from the iPad using the Gmail app or just sending it from within a Chrome window.

Thoughts?