how to check the port connectivity to remote server or within the server if telnet is not installed

Use Below command if Telnet is not installed in the server

Nice and verbose! From the man pages. 
Single port:
nc -zv 127.0.0.1 80
Multiple ports:
nc -zv 127.0.0.1 22 80 8080
Range of ports:
nc -zv 127.0.0.1 20-30
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you can us cat also to check the connectivity 
cat < /dev/tcp/127.0.0.1/22
SSH-2.0-OpenSSH_6.6.1