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## Optimizing parameters

I came across a fun problem recently that gave me a good opportunity to exercise my approximation muscles.

Problem: Compute $\displaystyle \lim_{n \to \infty} \frac{n + \sqrt{n} + \sqrt[3]{n} + ... + \sqrt[n]{n}}{n}$, if it exists.

The basic approach to such sums is that the first few terms contribute to the sum because they are large and the rest of the terms contribute to the sum because there are a lot of them, so it makes sense to approximate the two parts of the sum separately. This is an important idea, for example, in certain estimates in functional analysis.

Since $\sqrt[k]{n} \ge 1, k \ge 2$ it follows that the limit, if it exists, is at least $\lim_{n \to \infty} \frac{2n-1}{n} = 2$. In fact, this is the precise value of the limit. We’ll show this by giving progressively sharper estimates of the quantity

$\displaystyle E_n = \frac{1}{n} \sum_{k=2}^{n} \left( \sqrt[k]{n} - 1 \right)$.

In the discussion that follows I’m going to ignore a lot of error terms to simplify the computations.