The answer is No! No you can't do that. It's a breach of the SPAM act for a start and you could end up in real hot water, not to mention you will most likely annoy many of your connections and lose a lot of them! By adding your LinkedIn connections to your list is illegal because it is SPAM. Unless you have been given permission for someone’s email to be added to your list, you are violating the CAN-SPAM Act of 2003. So don't do it.
What prompted me to write this post is because a LinkedIn connection I've known personally for over 12 years, and is respected marketer, keeps emailing me, despite the fact I have unsubscribed many times. His emails are no longer relevant to my work and just another email I have to delete! I never read them so he's completely wasting his time. When I unsubscribed I noticed part of his campaign list was "LinkedIn Subscribers June 2016" - so that was it! No matter how many times I unsubscribe from is mailing list he keeps updating it by downloading his LinkedIn connections.
I immediately disconnected him as a LinkedIn contact.
If LinkedIn is your preferred social media, and I'll be honest with you, it's no longer mine, then there are other ways to use it to gain clients without adding people to your emailing list. Let's face it, unless your newsletters are "GOLD" full of information that your readers want then most likely they will get deleted anyway. What you can do is start using LinkedIn as a 'blog' and write interesting articles and post. You will be surprised how much interaction you will get if you share really good content.