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No rational function

Prove that there exists no rational function such that

\displaystyle f(n)=1+ \frac{1}{2} + \cdots + \frac{1}{n} \quad \text{forall} \; n \in \mathbb{N}

Solution

Suppose , on the contrary , that such function exists. Since the harmonic series diverges we conclude that the limit of our function in infinity is infinity. This, in return means that the degree of the nominator , call that m is greater that the one of the denominator , call that n. Extracting x^{n-m} in the nominator we get that

\displaystyle R(x)=\frac{P(x)}{Q(x)} = x^{m-n} s(x)

The limit of s at infinity is finite and call that \ell. Hence:

\begin{aligned} \lim_{x \rightarrow +\infty} \left ( \frac{f(x)}{g(x)} - \ln x \right ) &= \lim_{x \rightarrow +\infty} \left [ x^{m-n} s(x) - \ln x \right ] \\ &= \lim_{n \rightarrow +\infty} x^{m-n} \left [ s(x) - \frac{\ln x}{x^{m-n}} \right ]\\ &= +\infty \end{aligned}

and of course this contradicts the fact that

\displaystyle \lim \left ( \mathcal{H}_n - \ln n \right ) = \gamma

where \gamma is the Euler  – Mascheroni constant .

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