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

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?

Bạn đang xem: 1+1/2+1/3+...+1/n

1 + một nửa + 1/3 + 1/4 +.... + 1/n

asked Sep 23, 2019 at 17:26

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

There is no simple expression for it.

Xem thêm: tập thể tiêu biểu trong xây dựng, giữ gìn, phát huy giá trị truyền thống của llvt thủ đô hà nội

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

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