I think this can be widely different, depending on the type of series. If you have a series with a complete arc that spans multiple books in the series, then you will probably need to have some level of outline or synopsis written out.
If the series is more of an adventure (or whatever) in one book that is done, and then in book two you just start a new adventure, then not so much.
In my case, I have written book one of what COULD turn out to be a 4-book series. I have 4 books plotted out, and that completes the story I have to tell. There is a lot of leeway in that plot...but there are key plot elements that MUST be met in order for the entire story to work. Once book one is published, I can't go back and fix it should I find in, say, Book three I decide to go in a different direction. That's because certain things happen in book one (even though it isn't obvious to the reader) that tie to things that will happen in subsequent books.
For me, however, most of my "outline" is in my head. I have notes/ideas. But I know what is going to happen in each book well enough to not write it down. I will, however, write out an outline for each book as I reach it, should it ever come to that.