A student asked me how I'd suggest writing a "series synopsis," to go along with the proposal of the first book. (Now as Jami just reminded me, I must define that: Proposal usually equals the first 3 chapters of the first book, and a longer, more detailed synopsis of that first book. That is, this series synopsis accompanies, doesn't replace, the book synopsis.)
She wanted, sensibly, to give an idea of what the trilogy or tetraology is about, along with the more detailed first-book synopsis. But she didn't have much of the plot of the other books outlined.
I suggested that a series synopsis is (in my experience) about a page long, and goes along with the synopsis for the first book. It isn't about plot (of course, I'd say NO synopsis should be about plot-- plot-based synopses are usually so boring), but rather about the problem or situation that will carry through the whole series.
You outline that series-long situation and how it affects the main characters ("Jack Aubrey's ships are all involved in the naval war against Napoleon. This war rages over six seas and four continents, and involves complex historical issues like slavery, democracy, and colonialism. Jack is always in the middle of the action. But the huge war is not the only conflict he faces. Starting out as a young, merry lieutenant, he grows in every command to become a leader, but the problems he faces on each ship teach him to be a man too. ETC.") or the group. ("In The Church of England series, between-the-wars Anglican clergymen face dangers theological and personal as their country and their parishioners change around them during the decades leading to World War II. One by one, the clerics achieve power through their ambition, but are brought down by their spiritual failings. ETC.")
Then you might quickly sketch the first book in a paragraph, and then finish up with a paragraph about the rest:
"The second book (title) explores the growing tensions in the friendship between Stephen and Jack as they compete for the same woman. In the background of this very personal battle is the war with Napoleon, which rises to an explosive point when.... Other books in the series will address (a few big-picture and personal things). The last book of the series will resolve the major questions and reveal (some major revelation), finally bringing the warriors home to England as the war and its aftermath finally wind to a close."
That is, set up the situation that makes this a series, what connects all these books. Then sketch the first book. Then (quickly) sketch the rest of the series. You don't have to know every plot, just what the series as a whole will confront and eventually resolve.
What do you all think? Have you ever submitted a series? Did you do a series synopsis? What did the agent or editor want?