Fraction Calculator: 1/2 : 1/3

     
How bởi vì I calculate this sum in terms of "n"?I know this is a harmonic progression, but I can"t find how to calculate the summation of it. Also, is it an expansion of any giayphutyeuthuong.vnematical function?

1 + một nửa + 1/3 + 1/4 +.... + 1/n

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

$$sum_r=1^n frac1r approx int_1^n fracdxx = log n $$

So as a ball park estimate, you know that the sum is roughly $log n$. For more precise estimate you can refer to lớn Euler"s Constant.

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There is no simple expression for it.

But it is encountered so often that it is usually abbreviated to $H_n$ & known as the $n$-th Harmonic number.

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

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For example, $$H_n=G_n-(n+1)lfloorfracG_nn+1 floor$$ where $$G_n=fracn+(n+1)!choose n-1(n+1)!$$

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


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

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Of course, one reason for creating the digamma function is khổng lồ make formulaelike this true.


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how khổng lồ find a sum of numbers in a sequence when some intermediate terms are not taken in to consideration?
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