Take a story's temperature by studying its tone. Is it hopeful? Cynical? Snarky? Playful?
As our narrator, Jim Hawkins definitely has a flair for the dramatic. He likes to remind us that he's involved in a high-stakes search for treasure. The potential reward is enormous (gold bars) but the risks are equally intense. Take, for example, when Jim ends Chapter 10 with a description of his feelings upon overhearing Long John Silver's mutiny plot:
It was Silver's voice, and, before I had heard a dozen words, I would not have shown myself for all the world, but lay there, trembling and listening, in the extreme of fear and curiosity; for from these dozen words I understood that the lives of all the honest men aboard depended on me alone. (10.30)
It's the end of a chapter, and Jim wants us to keep reading, so he doesn't tell us exactly what Long John Silver's "dozen words" that so terrify Jim are. Instead, he focuses on the effect Long John Silver is having on Jim, leaving him "trembling [...] in the extreme of fear and curiosity." This intense description of Jim's fear combines with his deliberate withholding of information until the next chapter to make us intensely curious about what's going to happen next.
Jim's narration seems to be expressly designed to thrill, chill, and above all interest us in the events of the novel. While Doctor Livesey doesn't seem to have quite Jim's skill, he does keep the suspense going across the three chapters he narrates. For example, at the end of Chapter 18, he suddenly wonders what happened to Jim. Just as he states the question, Jim appears, "safe and sound, [coming] climbing over the stockade" (3.18.41). What has Jim been doing? How has he reached the fort? We have to keep reading to find out.