What are the defining characteristics of series novels?What value do formulaic books hold for their readers (especially children)? Examples include Captain Underpants and Goosebumps.
If we were looking at such works from a thematic point of view, most of these works embrace a form of the opportunity ideology. Essentially, this means that these stories depict characters who persevere through a sense of will and commitment to a higher end. There is not a massive questioning of existing social structures and a debate of how power is distributed in such settings. Rather, these works show characters who have a combination of natural talent and a willingness to "go the extra mile" in order to achieve what they want. These works feature characters who are able to summon a certain level of courage and internal strength to rise to an occasion and do great things.
To a certain extent, this makes sense to integrate this theme in children's literature. It might not allow children the hope to believe as much if their works were littered with the idea that unless they are in the richest 1% of the world's economic powers, they lack a sense of autonomy or control over their lives. Children usually do not immediately gravitate towards works that devalue the individual's efforts in the face of political or economic determinism. However, I still believe that it is important to depict works that honestly show this reality and combine how individuals still have power in the face of such conditions. Children's literature publishers might not entirely agree with this, though.
Book series are equivalent to TV series, an extremely popular form of entertainment. The characters and settings are consistent, of course, and the kinds of plots are ultimately rather predictable (i.e., friend drama, crime investigation). We know it works; the evidence is clear. On the negative side, many actors create such memorable characters that they're typecast for the rest of their careers (think Kramer on Seinfeld). On the positive side, popular series are often the source of spin-offs (think Law and Order or CSI, or Frasier). In the middle is the reality shows, in which real-life characters simply live their lives (Real Housewives) or where a revolving cast of characters repeat a formulaic structure (Survivor, The Bachelor). Book series are the same--we follow them because we generally develop relationship first with the characters, then with the action and structure.
One of the most important value of series novels is that intelligence will eventually will win out over strength, and that the good guy, in the end, will always win. Even at times where this might not be true, it is only a cliffhanger for the next book to come along.
The defining characteristics of series novels include recurring themes, narrative style, characters, and sometimes setting. For reluctant or new readers books such as the examples you gave which tend to be very formula driven are comforting because the reader knows what to expect and the vocabulary is generally less advanced. Adult versions of this exist and some writers have been very successful with "pulp fiction" such as Sue Grafton, Janet Evanovitch, J.D. Robb (Nora Roberts), and Dean Koontz.
For older readers series such as The Chronicles of Narnia, A Series of Unfortunate Events, Twilight, or Harry Potter provide more advanced plots and characters who are developed more completely over time. As mentioned above in other answers, one of the best things about a series is that if the first book provides a good experience the reader can go back to the same author and quite often once they complete the series they go looking for other similar books. I have had a lot of students who were "hooked" into reading by a series.