Did you just spell "to" as "2"? And what's a "maximum sum"? Anyway,

use List::Util qw( sum );
my $sum_of_pos = sum grep { $_ > 0 } @array;
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or

my $sum_of_pos = 0;
for (@array) {
   $sum_of_pos += $_ if $_ > 0;
}
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Update: Oops, missed the "continues". Maybe that's a misspelling of "contiguous", and maybe you mean "contiguous and increasing"???

my $prev;
my $sum;
my $max = -1;
for (@array) {
   if (defined($prev) && $_ == $prev+1) {
      $prev = $_;
      $sum += $_;
      next;
   }

   if (defined($prev)) {
      $max = $sum if $sum > $max;
      undef $prev;
      undef $sum;
   }

  #if ($_ >  0) {  # zero isn't considered positive
   if ($_ >= 0) {  # zero is    considered positive
      $prev = $_;
      $sum  = $_;
   }
}
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