"Sumimasen" is more formal, a way of apologizing to a stranger.If you use "gomenasai", it either sounds like you're being very childish or that you just did something that requires a sincere apology. "Gomen" on the other hand is more natural when speaking with a friend or family member. (^‿^✿)
So it seems like in many cases (and as I've read in comments here) that ません is used for a negative connotation while ます is used for a positive. Does this mean that there is a positive version of すみません? Would you ever say すみます? And if so, what is the translation for すみ in this context?
I found this while searching for an answer: "The etymology of "sumimasen" is interesting. During the Edo period, there was a lot of trouble over money among people; purchasing and paying, loaning and returning, and so forth and so on.
When everything was paid back and cleared, it was expressed by the verb "sumu" (done 済む). In other words it was finished, cleared.
When something remained in debt, it was "sumanu" or "sumanai", which became "sumimasen". Now, however, most situations involving sumimasen are not related to money trouble. Money trouble today cannot be remedied by merely saying "sumimasen"." source: http://languagelog.ldc.upenn.edu/nll/?p=9155
TL;DR: There is not a positive version of phrase すみません (sorry), but there is a positive version of the verb すみません (to not finish), that is to say, すみます (to finish).
Bonus: すみます, when spelled 住みます, means to reside in [a country, city, etc]. Ex. アメリカに住みます (I live in America)
すみません is also used to get someone’s attention, as in a store or boutique when you have a question or need help. You can use this to initiate contact. “すみません。。。” and you ask your question. Like すみません、これはいくらですか。You’ll generally do it with a bow, and even if they don’t understand or hear your “すみません”, a Japanese will always bow back out of reflex, and you can continue with your question or interruption.
Going off of たのしかった(tanoshikatta), it might denote that the apology is for something in the past. So, something along the lines of "I'm sorry about that", when talking about something that was in the past. Japanese tends to be context-bound anyway, afaik. Don't take me up a 100% on this through, since this is just going off of one thing to the other and thus might be wrong.
It's somewhat of an apologetic 'Thank you!' for something that another has gone to the trouble of doing for you. If somebody helps you find something you lost for example, you can use it. Saying どうもすみません pretty much intensifies the phrase and also, from the native usage I've seen, conveys explicitly that 'Thank you' nuance.
Duolingo, this is not accurate in the way that Japanese people use this word. Sumimasen is much more, excuse my interruption or excuse some slight offense that is necessary. Gomenasai is "Sorry!" much more aligned with Japanese. PLEASE fix ro remove. It is teaching the wrong usage!
This is one of the few Japanese words that I recognize- I only know what it means because me and my family would watch James May: Our Man In Japan and he would turn to the camera before every activity and say it as an, "I'm sorry you have to see this." because he was so bad at every activity.