# 1+1/2+1/3+...+1/n

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How bởi I calculate this sum in terms of 'n'? I know this is a harmonic progression, but I can't find how vĩ đại calculate the summation of it. Also, is it an expansion of any mathematical function?

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1 + một nửa + 1/3 + 1/4 +.... + 1/n


asked Sep 23, 2019 at 17:26 $\endgroup$

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There is no simple closed khuông. But a rough estimate is given by

$$\sum_{r=1}^n \frac{1}{r} \approx \int_{1}^n \frac{dx}{x} = \log n$$

So as a ball park estimate, you know that the sum is roughly $\log n$. For more precise estimate you can refer vĩ đại Euler's Constant.

answered Sep 23, 2019 at 17:41 $\endgroup$

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There is no simple expression for it.

But it is encountered so sánh often that it is usually abbreviated vĩ đại $H_n$ and known as the $n$-th Harmonic number.

There are various approximations and other relations which you can find in Wikipedia under Harmonic Number or in the question Jose Santos referenced in the comments.

For example, $$H_n=G_n-(n+1)\lfloor\frac{G_n}{n+1}\rfloor$$ where $$G_n=\frac{{n+(n+1)!\choose n}-1}{(n+1)!}$$

But that kind of thing is more of a curiosity than vãn a useful expression!

answered Sep 23, 2019 at 17:32 $\endgroup$

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One can write $$1+\frac12+\frac13+\cdots+\frac1n=\gamma+\psi(n+1)$$ where $\gamma$ is Euler's constant and $\psi$ is the digamma function.

Of course, one reason for creating the digamma function is vĩ đại make formulae like this true.

answered Sep 23, 2019 at 18:22 $\endgroup$