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?

#Calculus #HarmonicSeries #Harmonic #Growth

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Comments

Somewhat relevant.

Pi Han Goh - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$