Find the least integer $k$ such that $left[H_kright]=n$ for a given positive integer $n$
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At the moment I am curious about harmonic numbers at their properties and came across the following question.
We are given $ninmathbb{N}$. We are to find the least integer $k$ such that $left[H_kright]=n$ where $H_n$ is the $n$-th harmonic number and $[cdot]$ is the integer part.
My attempt. It is known that $H_n=gamma+psi(n+1)$ where $psi(x)$ is digamma function. So I got $$left[H_kright]=n\nleq H_k<n+1\nleq gamma+psi(k+1)<n+1\n-gammaleq psi(k+1)<n-gamma+1\psi^{(-1)}(n-gamma)-1leq k<psi^{(-1)}(n-gamma+1)-1
$$
where $psi^{(-1)}(x)$ is inverse function of $psi(x)$. $psi(x)$ is increasing for $x>0$.
And I stuck here. I do not know how to show that there exists integer $k$ in interval $left[psi^{(-1)}(n-gamma)-1;psi^{(-1)}(n-gamma)-1right)$ at all.
I suppose that I've chosen too complicated way to tackle this problem. I appreciate any hint or answer. Thanks!
harmonic-numbers digamma-function
asked Nov 22 at 10:17
Mikalai Parshutsich
395214
add a comment |
up vote
0
down vote
favorite
At the moment I am curious about harmonic numbers at their properties and came across the following question.
We are given $ninmathbb{N}$. We are to find the least integer $k$ such that $left[H_kright]=n$ where $H_n$ is the $n$-th harmonic number and $[cdot]$ is the integer part.
My attempt. It is known that $H_n=gamma+psi(n+1)$ where $psi(x)$ is digamma function. So I got $$left[H_kright]=n\nleq H_k<n+1\nleq gamma+psi(k+1)<n+1\n-gammaleq psi(k+1)<n-gamma+1\psi^{(-1)}(n-gamma)-1leq k<psi^{(-1)}(n-gamma+1)-1
$$
where $psi^{(-1)}(x)$ is inverse function of $psi(x)$. $psi(x)$ is increasing for $x>0$.
And I stuck here. I do not know how to show that there exists integer $k$ in interval $left[psi^{(-1)}(n-gamma)-1;psi^{(-1)}(n-gamma)-1right)$ at all.
I suppose that I've chosen too complicated way to tackle this problem. I appreciate any hint or answer. Thanks!
harmonic-numbers digamma-function
asked Nov 22 at 10:17
Mikalai Parshutsich
395214
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
At the moment I am curious about harmonic numbers at their properties and came across the following question.
We are given $ninmathbb{N}$. We are to find the least integer $k$ such that $left[H_kright]=n$ where $H_n$ is the $n$-th harmonic number and $[cdot]$ is the integer part.
My attempt. It is known that $H_n=gamma+psi(n+1)$ where $psi(x)$ is digamma function. So I got $$left[H_kright]=n\nleq H_k<n+1\nleq gamma+psi(k+1)<n+1\n-gammaleq psi(k+1)<n-gamma+1\psi^{(-1)}(n-gamma)-1leq k<psi^{(-1)}(n-gamma+1)-1
$$
where $psi^{(-1)}(x)$ is inverse function of $psi(x)$. $psi(x)$ is increasing for $x>0$.
And I stuck here. I do not know how to show that there exists integer $k$ in interval $left[psi^{(-1)}(n-gamma)-1;psi^{(-1)}(n-gamma)-1right)$ at all.
I suppose that I've chosen too complicated way to tackle this problem. I appreciate any hint or answer. Thanks!
harmonic-numbers digamma-function
asked Nov 22 at 10:17
Mikalai Parshutsich
395214
At the moment I am curious about harmonic numbers at their properties and came across the following question.
We are given $ninmathbb{N}$. We are to find the least integer $k$ such that $left[H_kright]=n$ where $H_n$ is the $n$-th harmonic number and $[cdot]$ is the integer part.
My attempt. It is known that $H_n=gamma+psi(n+1)$ where $psi(x)$ is digamma function. So I got $$left[H_kright]=n\nleq H_k<n+1\nleq gamma+psi(k+1)<n+1\n-gammaleq psi(k+1)<n-gamma+1\psi^{(-1)}(n-gamma)-1leq k<psi^{(-1)}(n-gamma+1)-1
$$
where $psi^{(-1)}(x)$ is inverse function of $psi(x)$. $psi(x)$ is increasing for $x>0$.
And I stuck here. I do not know how to show that there exists integer $k$ in interval $left[psi^{(-1)}(n-gamma)-1;psi^{(-1)}(n-gamma)-1right)$ at all.
I suppose that I've chosen too complicated way to tackle this problem. I appreciate any hint or answer. Thanks!
harmonic-numbers digamma-function
harmonic-numbers digamma-function
asked Nov 22 at 10:17
Mikalai Parshutsich
395214
asked Nov 22 at 10:17
Mikalai Parshutsich
395214
asked Nov 22 at 10:17
Mikalai Parshutsich
395214
asked Nov 22 at 10:17
Mikalai Parshutsich
395214
asked Nov 22 at 10:17
Mikalai Parshutsich
395214
395214
add a comment |
add a comment |
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