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Review of Applied Thermodynamics for Engineering Technologists. PDFÂ Solution Manual and eBook.
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Solution Manual and eBook.Q:

How to show $\sum_{i=1}^n \frac{1}{n} + \frac{1}{n} +… + \frac{1}{n} \le \ln(n)$

How do I show that, for $n \in \mathbb{N}^+$,
$$\sum_{i=1}^n \frac{1}{n} + \frac{1}{n} +… + \frac{1}{n} \le \ln(n)$$
or at least that
$$\sum_{i=1}^n \frac{1}{n} \le \ln(n)$$
It is clear that the right hand side approaches $\ln(n)$, but I’ve no clue how to show the inequality.

A:

Note:
$$\sum_{i=1}^n \frac{1}{n} + \frac{1}{n} +… + \frac{1}{n} = \sum_{i=1}^n \frac{i}{n} = \sum_{i=1}^n \ln i \le n \ln n$$

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