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### Why do we say the harmonic series is divergent? [duplicate]

If we have $\Sigma\frac{1}{n}$, why do we say it is divergent? Yes, it is constantly increasing, but after a certain point, $n$ will be so large that we will be certain of millions of digits. If we ...
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### Is the sequence of sums of inverse of natural numbers bounded? [duplicate]

I'm reading through Spivak Ch.22 (Infinite Sequences) right now. He mentioned in the written portion that it's often not a trivial matter to determine the boundedness of sequences. With that in mind, ...
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### Prove sequence $H_n=1+\frac{1}{2}+\frac{1} {3}+\dots+\frac{1}{n}$ does not converge. [duplicate]

I want to prove that the harmonic series $H_n=1+\frac{1}{2}+\frac{1} {3}+\dots+\frac{1}{n}$ does not converge. My attempt: to see it does not converge, I try to prove that it is unbounded and it is ...