Notes - I used the Vendor/Banner method and nothing appeared except a lot of blank space (which was the same result as copy/paste in the Settings header/footer of the HTML module.
- After setting up an Adsense account, it requires a waiting period (account review - not like buying a firearm or anything, but it does take time

)...
During this 'time' the ads are blank but after this 'time' has passed email confirmation letting you know you've been approved, the adsense ads start to appear.
I have implemented this before, but it was in an early 5.x version when there was an Adsense module and much of the guess work was removed.
Last note, make sure the page with the banner or HTML module is visible to the public for the review 'time' (I didn't list this page as a large blank area didn't seem all that interesting).