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**ThePerfectHacker** The sum of reciprocals $\displaystyle \sum_{n=1} \frac{1}{n}$ is called the "Harmonic Series". Note, $\displaystyle \lim \frac{1}{n}=0$. However, the theorem does not say whether it converges or diverges, thus we do not know. But soon we shall.

Let us determine whether $\displaystyle \sum_{k=1}^{\infty} \frac{1}{n}$ converges or diverges. Note an extension function is $\displaystyle f(x)=1/x$. This function is continous, positive, and decreasing (the derivative is negative). Thus, we can compare it with $\displaystyle \int_1^{\infty} \frac{1}{x} dx = \lim_{t\to \infty} \ln t -\ln 1 = \lim_{t\to\infty} \ln t$. This grows without bound. Thus, the harmonic series diverges.