I'll not comment on the errors in the algorithm - after all, it's your homework and not mine. So, here are some Perl tips:

• Read your variable like this:

  print "Please enter a number: \n";
  my $limit= <STDIN>;
  chomp $limit;
  [download]

  Note there are no parentheses around $limit in the declaration. With the parentheses, you enter "list context" - which means you'll read line after line from STDIN, until you hit Ctrl-D for "end of file".
• Every statement ends with a semicolon; this is missing in your line $sum += 2. That's why you get a syntax error.
• If your output should contain the phrase (1+3+5), then take a look at the join function. It should give you a hint what to push into your @array.