"I thank you."
Translation:Jeg takker dig.
Never heard "jeg takker dig" from danes, neither "i thank you" from English speakers. What about "tak", "mange tak", "tak skal du have"? Let's use real phrases
I've definitely heard "I thank you" from English (note: in the U.S.) speakers (e.g. "I thank you all for coming today"). It's only uncommon outside of formal contexts.
As a native speaker I'd have to agree. The only time where I would think this phrase is natural is in writing, like "Du giver mig en gave. Jeg takker dig og nyder bagefter juleaften med familien" ("You give me a present. I thank you and then enjoy Christmas Eve with the family")
I.e. not something you would say to someone's face, but rather read in a book written in present tense.
"I thank you" is actually used a lot in the US. It was mostly used in old English. It is more of an old politeness that is still in some use today. I use it every once in a while myself.
"Du" can only be used when "you" is the one doing the action. "Dig" is used when "you" is the one the action is being performed to.
Compare to the English "I" and "me" in the first person ("jeg" and "mig" in Danish respectively). "Du" is the second person equivalent of "I" and "dig" is the second person equivalent of "me".