I don't speak Ruby at all, and the Sinatra site isn't terribly clear on what it is (is it a framework for Ruby?) so I might be completely off my rocker here, but:
Connection: close
is sent by your Web server when keep alives are turned off. For scalability reasons, keep alives are generally considered to be step one on things to turn off in your server. To be fair, there's a school of thought both ways, particularly when Ajax is involved.
I use nginx for my Django work (I'm thinking it's similar), and I have keep-alives turned off in nginx like this:
14:58 jsmith@lateralus% grep alive /etc/nginx/nginx.conf ~
keepalive_timeout 0;
Apache uses KeepAlive (see here).
If Sinatra is its own Web server, I can't find any documentation to turn keep alives on, and I'll go ahead and eat the fact that I look like an idiot.