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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
2 years, 1 month ago

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