What I want from a series of novels is to know who this series will feature. If it's a hero, i.e. Mitch Rapp, then I want him to be interesting enough to carry however many books are written about him. Same with a heroine. If the series features a family, each member getting the role of main character interacting with the other members to be revealed later, I want to know what to expect in the future.
With long-running series it's not easy to keep the hero in constant focus. Authors of my favorite series have veered off with strong peripheral characters to change it up once in a while. The hero is present in the story and is either a friend or an enemy to the character, but his presence is diluted until a critical occurrence. I admit those are usually my least favorite stories in the series - especially if the hero/main character is a stand-out and he's the reason you're buying the book.
When family series are written, an author must make each member a potent enough character to carry the reader to the next family member. That's not an easy task. Readers form favorites quickly - I know I do - and might become unattached to the next in line. Just as authors often love a challenge, so do readers. If a reader likes that first story, they'll follow up until one of the characters disappoints or bores them.
As an author, I haven't considered doing a series although I've heard a few requests for one. As a reader, my devotion to series-reads lies mostly within the thriller genre where several impressive heroes reign. Top of that list is the hero Mitch Rapp. No hero has yet to topple Rapp from the top of the pile. Doubtful anyone will.
I want my heroes and heroines to be strong enough to carry the load of series novels, interesting enough to reveal nuances of who they are as they continue to grow, and to give me that certain something - whatever it is - that keeps them separate from the rest and unique enough to remain there.
Father, I continue to thank you for your writers/authors. Thank you for allowing me to write. Direct me, my words and stories. May I always honor you with what you've given me. In the Name of Jesus, Amen.