This week's section is called Vayechi. The main topics are Jacob's blessing on his sons and his passing. On the latter point, our sages dwell on the peculiar language used by the Torah to describe Jacob's death (Genesis 49:33): "[...] he breathed his last, and joined his people." The verse never actually states that Jacob died... from this they derive the idea that Jacob, our patriarch, did not pass away.
Obviously, the statement is controversial, primarily because the Torah itself recounts the process following his death, when his body was prepared and even buried in Mearat haMachpelah, where the other patriarchs were buried. Therefore, our sages conclude that, just as his descendants (the Jewish people) are alive, he (Jacob) is also alive.
We find the same idea regarding King David when we say (in the Siddur, Kiddush Levanah), "David, the King of Israel, is alive and eternal." Now, regarding David we say that "he is alive," but regarding Jacob we say that "he did not die." What is the difference?
Perhaps we could say that being alive means that a person has a revealed vitality, is aware of their own being, and, so to speak, exercises their capacity to be alive. On the other hand, not being dead means that a person has a potential vitality, which is not expressed. Being alive is something recognizable to an external observer, while not being dead is something that is not visible from the outside.
Based on this idea, perhaps we could explain the difference between Jacob and David, which led our sages to express their "eternity" in different ways. David represents the glory of the Jewish people, their strength and beauty. Jacob (particularly this name, compared to the name Israel, which denotes a high spiritual level) represents each individual Jew, living the hardships of daily life, without beauty, without reign, without glory.
Therefore, when we speak of David, of the kingship of the Jewish people, this is something revealed to all peoples in different forms and expressions, whether in science, wisdom, history, values, etc. Thus, our sages say, "He is alive and eternal." When we speak of Jacob, the one who strives daily to eat kosher, to observe Shabbat, to study Torah, and to maintain peace in the home, without neglecting his work and the education of his children, our sages say, "He did not die.".
In other words, being part of the Jewish people gives a person a revealed and clear framework, one that comes "from above": "royalty and glory." In turn, it gives a person a hidden capacity, which it is up to each individual to reveal and utilize. It depends on personal, daily effort to move from a state of "not dead" to "alive and eternal," revealing that glory and beauty in each of the mitzvot of the Torah.