But such a demonstrations wasn't necessary for me since I had finished writing the following from scratch before you posted your solution. (I didn't post it since I didn't want to do the OP's homework for him.)

sub ikegami_exp {
   my ($x) = @_;
   my $last   = 0;
   my $result = 1;
   my $acc    = 1;
   for (my $n = 1; abs($result-$last)>0; $n++) {
      $last = $result;
      $acc *= $x/$n;
      $result += $acc;
   }    
   return $result;
}

for (1, 0.1, 10, 3.14, 0.1234, 100) {
   printf("%.16e: exp() = %.16e  ikegami_exp() = %.16e\n",
      $_,
      exp($_),
      ikegami_exp($_),
   );
}
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The only difference was the terminating condition, so naturally, that's what interested me :) Mine goes into an infinite loop if you try to find ikegami_exp(1000), but that's easily fixed by replacing abs($result-$last)>0 with $result != $last.