Why do we stay up all night on Hoshanah Rabbah, and what is its source?
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Taamei Haminhagim (citing Machazeh Avraham, by R. Avraham of Buczacz) says that it is because the ushpiza for the seventh day of Sukkos is King David, and he used to stay up all night studying Torah - so we do the same to evoke his corresponding Divine attribute.
R. Chaim Vital (Pri Eitz Chayim and Shaar Hakavanos, cited in Nitei Gavriel) give a different reason: because it is the last night before the final "sealing" of our judgment for the year.