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How Slow Does Harmonic Series Grow?

Ok so we know that Harmonic series \(\displaystyle H_n = \sum_{k=1}^n\dfrac1k\) diverges. And by integral test, we can see that \[\ln(n+1) = \int_{1}^{n+1} \dfrac{\mathrm d x}x ~~~<~~~ H_n ~~~<~~~ 1+\int_1^n\dfrac{\mathrm d x}x = 1+\ln(n)\]

Saying we use a computer that add a million of terms per second, and we have it doing that for a million years.

That would be \(n = 60^2\cdot24\cdot365\cdot10^{12} < 3.2\cdot10^{19}\)

Hence \(\ln(n) < \ln(3.2) + 19\ln(10) < 45\). Since \(H_n < 1 + \ln(n)\), that means after million years of addition, \(H_n\) still hasn't reached \(46\). Wow!

Isn't it amazing that \(H_n\) grows extremely slow, yet it diverges?

Note by Micah Wood
1 year, 4 months ago

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Somewhat relevant. Pi Han Goh · 1 year, 4 months ago

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