Final answer:
The narrator travels alone to the summit of Montanvert to test his physical strength and character.
Step-by-step explanation:
The reason why the narrator travels to the summit of Montanvert alone is because he wishes to prove his physical strength and character. In the story, the narrator is described as someone who enjoys solitude and seeks challenges to overcome. By climbing the summit alone, he tests his own abilities and proves his self-reliance.