As this infinite series could get rather large you may like to create an iterator to return the next member from the series each time it is called

#!/usr/bin/perl
use strict;
use warnings;

# make our iterator (a subroutine which holds its own state)
my $ones = seq_gen(1);

while (1) {
    # call iterator and print result
    my $next = $ones->();
    print "$next\n"; 
    sleep 1;
}

sub seq_gen {
    # generate and return a subroutine that captures its entry values 
+and
    # gives you the next member each time it is called
    my $seq = shift;
    my $string = '';
    return sub {
        $string .= $seq;
        return $string;
    }
}
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Cheers,
R.