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# [Jelly], 8 bytes

    İ€S<¬ʋ1#

[Try it online!][TIO-kqb1zm6r]

[Jelly]: https://github.com/DennisMitchell/jelly
[TIO-kqb1zm6r]: https://tio.run/##y0rNyan8///IhkdNa4JtDq051W2o/N9EwfpwuwpQRMH9PwA "Jelly – Try It Online"

## How it works

    İ€S<¬ʋ1# - Main link. Takes n on the left
         ʋ   - Group the previous 4 links into a dyad f(k, n):
     €       -   Over each integer 1 to k:
    İ        -     Get its reciprocal
      S      -   Sum
       <     -   Less than n?
        ¬    -   Logical not
          1# - Find the first integer k ≥ n such that f(k, n) is true