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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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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What have you tried?

You could:

use strict;
use warnings;

my @array = (1, 2, 3, 4, -1, -2, -3, 11, 12, 13);
my $max = 0;
my $sum;

    ($a = ($_ > 0 ... $b + 1 != $_)),
    ($a and $a == 1 and $sum = 0),
    ($a and $a !~ /E0/ and $sum += $b = $_),
    ($a and $sum > $max and $max = $sum)
    for @array;

print $max;
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which prints 36, but your teacher may not like that.

See what teach makes of this :)

Update: Corrected logic from 'add if larger than last value' to 'add if larger than last value added.

#! perl -slw
use strict;
 use List::Util qw[ reduce ];;

my @array=(1,2,3,4,-1,-2,-3,11,12,13);;
my $sum = @{
    reduce{
        @{ $a } = ( $a->[0]+$b, $b ) if $b > $a->[1];
        $a;
    } [0, -9e99 ], @array
}[0];

print $sum;
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Thank you. it's working fine.

I'm sure it is working fine, until your teacher realizes how advanced the technique is for your level and googles it. Your academic career worth it?

my @array=(1,2,3,4,-1,-2,-3,11,12,13);
my @sums=();
my $last=0;
map {
  push @sums, $last+$_ if $last>0 and $_>0;
  $last=$_;
} @array;
my $max=0;
map {$max=$_ if $_>$max } @sums;

print "Max sum: $max\n";
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