Rearrange the parts of the sentence in correct order.
Libraries' collections are
P: such as economic development, educational
Q: research and recreational pursuits
R: crucial for diverse purposes

Let's look at the links:
The introductory segment "Libraries' collections are" needs a predicate adjective or descriptor. Part R ("crucial for diverse purposes") perfectly completes this core thought.

Part P introduces examples of these purposes ("such as economic development, educational...").

Part Q lists the remaining noun phrases ("research and recreational pursuits") completing the list started in P.
Thus, the coherent sequence is RPQ. Full sentence: "Libraries' collections are crucial for diverse purposes such as economic development, educational research and recreational pursuits."