Your facebook profile page and business fanpage are not connected. Yes, a person can see your fanpage without being logged on or even on FB, BUT they do have to have an account with FB to become a fan.
If you want to invite people within an email outside of FB, you can invite them with a link to your business fanpage. Some things to keep in mind:
1) If you have less than 25 ppl currently on your fanpag, you will be given a long url link. To add a more professional touch for your link, use a url shortner like tinyurl.com. Copy & paste the link within the tinyurl box and they will give you a special url link that you can use as many times as you want to send out an invite to join your fanpage
2) Once you reach over 25 fans, go to
www.facebook.com/username and you can create a unique url for your fanpage (and profile page for that matter)
Some other tricks- add a fanpage badge to your website or blog to bring more traffic to your page as well.
To invite ppl that you are friends with on FB, you can just send them an invite message through your fan page.
Hope this helps!
Natasha