What a weird way to write the harmonic average.

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Write v_i = Var[X_i]. John writes

    t_i = \frac{\prod_{j\ne i} v_j}{\sum_{k=1}^n \prod_{j\ne k} v_j}.

But if you multiply top and bottom by (1 / \prod_{m=1}^n v_m), you just get

   t_i = \frac{1/v_i}{\sum_{k=1}^n 1/v_k}.

No need to compute elementary symmetric polynomials.

If you plug those optimal (t_i) back into the variance, you get

    \min Var[\sum t_i X_i] = 1/(\sum_{k=1}^n 1/v_k) = H/n,

where `H = n / (\sum_{k=1}^n 1/v_k)` is the Harmonic Mean of the variances.