That depends on the kind of series you are writing.
1. For over a hundred years, mystery writers in particular have written series based on what some call the continuing character. Sherlock Holmes remained the same as he went from separate adventure to separate adventure. Doctor Moriarty showed up in several adventures along the way, so there was a continuing antagonist in the series as well.
Television picked up on this kind of continuing character idea. Other genres like SF & F have also, particularly recently in the vampire/demon slayer type stories.
2. Although there are earlier examples (E E “Doc” Smith’s Lensmen series for example), it was really Tolkien’s publisher who coined the phrase “Trilogy.” That is a different kind of series, where each story/book has some independence and resolution, but where the larger “trouble” remains unresolved until the last book. Think Star Wars…but now with the likes of Harry Potter, we have gone beyond the mere “Trilogy” concept.
The bottom line is you need to end your story/book in whatever way you are most comfortable and satisfied, and don’t let anyone tell you it must be this way or that (except maybe your publisher). Still, I would urge you to consider your readers. Will they be satisfied with the story while at the same time wanting to read the next one?
It is a fine line we all have to guess at. If you see a massive sales drop between book one and book two, my guess is the reader did not get a good read for their money but found book one only a set-up to force them to buy book two, and that can tick people off… Not a good idea.
My recommendation, for what it is worth is to foreshadow all you want and leave unresolved whatever grand story might need to continue, but let each book be a book – a story with a beginning, middle and end all unto itself.
Now, how much of book one needs to go into book two – what my grandfather used to call (re: television) “exciting scenes from last week?” This is also a fine line each writer needs to walk. It is an art form. Too much intro., and people wonder why they bothered to read the first book. Too little, and you will lose any new readers you might otherwise interest.
At some point, Conan-Doyle had to assume people knew the Holmes character well enough for him to jump right into the story. Imagine if Spiderman had to review his origin every episode. On the other hand, imagine reading The Two Towers (Lord of the Rings) with all mention of Sauron, the essence of the ring, Mordor and the impending doom of Middle Earth removed. There is a lesson there, I think. It doesn’t have to be all bunched up at the beginning of book two like some 50-100 page prologue… You can save it for a need to know basis – and then backtrack to find the best place to insert the information… Just some thoughts.
In fact, next time I may blog on prologues…