Jelly, 3 bytes

İ€S 

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Goes over 5 significant figures, and does not use a builtin for Σ{1/1, 1/2, 1/3, ..., 1/n}. Rather, it uses builtins for {1, 2, 3, ..., n}, 1/n, and Σ.

Explanation:

݀S Main link, monadic ݀ 1 / each one of [1..n] S Sum of 

Jelly, 3 bytes

İ€S 

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1-indexed.

Explanation:

݀S Main link, monadic ݀ 1 / each one of [1..n] S Sum of 

Jelly, 3 bytes

İ€S 

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Goes over 10 decimals (~16), and does not use a builtin for Σ{1/1, 1/2, 1/3, ..., 1/n}. Rather, it uses builtins for {1, 2, 3, ..., n}, 1/n, and Σ.

Jelly, 3 bytes

İ€S 

Try it online!

Goes over 5 significant figures, and does not use a builtin for Σ{1/1, 1/2, 1/3, ..., 1/n}. Rather, it uses builtins for {1, 2, 3, ..., n}, 1/n, and Σ.

Explanation:

݀S Main link, monadic ݀ 1 / each one of [1..n] S Sum of