First off, I should define a Series Bible. It is a folder of files or a notebook or binder or whatever you use containing:
Backstory of character
Anything else useful for your series.
Mine also covers themes of the work, genre, research bibliography and marketing ideas since I need somewhere to put that stuff.
Here are the FIVE REASONS why you need a Series Bible:
1) It keeps characters consistent between multiple volumes and shows you how you are going to move the character through their overall arc. It helps you not kill a character or otherwise write them into an unbreakable situation when you need this character later in the series. (This happened to me during Faminelands. As this was my first series, I didn't use a series bible, but Orin's original fate was much different...)
2) It easily adds depth. In the next Chronicles of The Martlet, The Morality of the Necromancer, one of the plots is Eohan and Kian seeking out their Pa since they did not know what happened to him. The Series Bible let me know where they needed to go, which providence they originally came from, and the Great Houses they must interact with to get there--and how those Great Houses might interact with commoners who are receiving a nobleborn's education and which Great Houses are allies with House Eyreid.
3) It allows the setting to grow over the course of a series, rather than slamming the reader with all your ideas in the first book just because you think it's cool.
4) It helps eliminate those strange, one-off books in the series which make the reader scratch their head and think WTF?
While I loved the first four books in Tanya Huff's Valor Confederation, the fifth made me wonder why it was written. (BTW Tanya Huff is one of my FAVORITE authors and I highly recommend her science fiction and fantasy books.)
5) It helps you consistently follow the rules you set up for their world. It can eliminate unrealistic technology or magic exploits or accidentally having characters ignore technology that can get them out of whatever pickle they are in. I think this is imperative to an in-depth Science Fiction or Fantasy Series.
6) It helps to smooth the time slur between books. Does one book leave off immediately where the last book finished or is there a gap? The bible can tell you what happened in that gap and how will you inform the reader.
7) Finally it will help your characters not break any laws which you set up as important to them and their culture. Or if they do break laws for the plot the punishments will be consistent with the story you are telling.
Now here is the ONE REASON you might not want to take the time to build a series bible. It takes time and most of what is in the bible will not make it to the text of your novel.
For one off projects, such as The Grove, I have a single word document with names, character descriptions and place names, but I don't even bother writing a larger bible because of the time it takes. I have seen too many would-be authors in writing workshops writing and rewriting their character bibles or series bibles with nothing to show for their work. I believe we should be moving forward.
MY ADVICE is, if you have a multiple book series in mind, take the time to write the bible, but don't take too long on it. Mine normally takes me a week and then I add stuff as I write the first book in the series. For later books, if the protagonists are jouneying to new places or meeting new characters, then into the bible they go.
As always, good writing to you...