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

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!

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.

Not the answer you're looking for? Browse other questions tagged summation or ask your own question.

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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