What will be the summation formula for harmonic number with extra parameter?

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sum = n/1 + n/2 + ... ... ... + n/i

What is the formula that could be used to calculate sum?

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For example $$n\left(\ln{n}+\gamma+\frac{1}{2n}-\sum_{i=1}^{+\infty}\frac{B_{2i}}{2in^{2i}}\right),$$ where $B_{2i}$ they are Bernoulli numbers.