Why $\sum_{k=1}^n 1/k = \ln(n)$?

Asked May 18 '21 at 10:18

Active May 18 '21 at 11:03

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In the book Introduction to Algorithms (3rd edition) I found this equation in a probability topic:

However, when I sum 1/1 + 1/2 + 1/3 + 1/4 + 1/5 it doesn't give me the same result as ln(5)

edited May 18 '21 at 11:03

Gary

asked May 18 '21 at 10:18

GabrielBB

3

As you can see, there is an error term of order $1$ there as well. See my answer here: https://math.stackexchange.com/q/3551025 You can also Google for Euler-Mascheroni constant, or maybe look at equation (A.7) in the book you are citing. – Gary May 18 '21 at 10:20
5

You are ignoring the $O(1)$ term, which represents the error. The sum is a Riemann Sum for $\int_1^x\frac {dt}t=\ln(x)$. – lulu May 18 '21 at 10:21
1

This may help: https://math.stackexchange.com/a/3960344/752069 – VIVID May 18 '21 at 10:30
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See also here. – J.G. May 18 '21 at 11:39

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