Today I would like to share another amazing property (convergence) of Digamma Functions. I found this one while playing with some integrals and their series expansion :
For all \(x > 0\)
\[\lim_{n \rightarrow \infty} \psi(nx+n+1)-\psi(nx+1) = \log \left(1+\frac{1}{x}\right)\]
Proof.
Great work by my friends, Ronak and Pratik.
So, here' my method.
Proof 1
We'll compute the following sum in two different ways \[\lim_{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{nx+r}\]
We can convert the above riemann sum into integral as :
\[\lim_{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{nx+r} = \int_0^1 \frac{dt}{x+t} = \log\left(x+t\right)|_0^1 = \log\left(1+\frac{1}{x}\right)\]
Also, we can compute the above sum by noticing that \[\frac{1}{A} = \int_0^\infty e^{-Ax}\ dx\]
Thus \[\begin{eqnarray} \lim_{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{nx+r} & = & \lim_{n \rightarrow \infty} \sum_{r=1}^n \int_0^\infty e^{-(nx+r)t}\ dt \\ \end{eqnarray}\] Because the integral is independent of the sum, we can interchange the sum and integral as \[\begin{eqnarray} \lim_{n \rightarrow \infty} \sum_{r=1}^n \int_0^\infty e^{-(nx+r)t}\ dt & = & \lim_{n \rightarrow \infty} \int_0^\infty \sum_{r=1}^n (e^{-t})^{r} e^{-nxt}\ dt \\ \end{eqnarray}\] Setting \(y = e^{-t}\) yields, \[\begin{eqnarray} \lim_{n \rightarrow \infty} \int_0^\infty \sum_{r=1}^n (e^{-t})^{r} e^{-nxt}\ dt & = & \lim_{n \rightarrow \infty} \int_0^1 \sum_{r=1}^n y^r\ y^{nx}\ \frac{dy}{y} \\ & = & \lim_{n \rightarrow \infty} \int_0^1 \frac{y^n-1}{y-1} y^{nx}\ dy \\ & = & \lim_{n \rightarrow \infty} \int_0^1 \frac{y^{nx+n}-y^{nx}}{y-1}\ dy \\ \end{eqnarray}\] Separate the above sum as \[\begin{eqnarray} \lim_{n \rightarrow \infty} \int_0^1 \frac{y^{nx+n}-y^{nx}}{y-1}\ dy = \lim_{n \rightarrow \infty} \left( \int_0^1 \frac{y^{nx+n}-1}{y-1}\ dy - \int_0^1 \frac{y^{nx}-1}{y-1}\ dy \right) \\ \end{eqnarray}\] Now, remember that by the definition of digamma function, we have \[\psi(s+1) = -\gamma + \int_0^1 \frac{t^s-1}{t-1}\ dt\]
Thus, we can conclude that \[\lim_{n \rightarrow \infty} \sum_{r=1}^n \frac{1}{nx+r} = \lim_{n \rightarrow \infty} \psi(nx+n+1) - \psi(nx+1) = \log\left( 1+\frac{1}{x}\right)\]
Proof 2
My other proof was exactly same as Ronak's, which makes use of recurrence function of digamma function.
Furthermore, I make a conjecture here that the above property holds true for some complex \(x\) too but I'm still working on it's proof. Any help would be appreciated.
Thanks,
Kishlaya Jaiswal.
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Kishlaya Jaiswal
3 years, 2 months ago
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Top NewestHere's what I did :
As we know, \(\psi(p+1)=H_{p}-\gamma\).
Also, \(H_{p}=\displaystyle\sum_{r=1}^{p} \dfrac{1}{r}=\displaystyle\int_{0}^{1}\dfrac{x^p-1}{x-1}\mathrm{d}x\).
So making use of the above equalities, our expression becomes \[\begin{align} \psi(nx+n+1)-\psi(nx+1)&=\lim_{n \to \infty}\int_{0}^{1} \dfrac{t^{nx+n}-1}{t-1}\mathrm{d}t-\int_{0}^{1} \dfrac{t^{nx}-1}{t-1}\mathrm{d}t\\ &=\lim_{n \to \infty} \int_{0}^{1}t^{nx-1}\dfrac{t(t^n-1)}{t-1}\mathrm{d}t\\ &=\lim_{n \to \infty}\int_{0}^{1}t^{nx-1}\sum_{r=1}^{n}t^r \mathrm{d}t\\ &= \lim_{n \to \infty}\sum_{r=1}^{n}\int_{0}^{1}t^{nx+r-1}\mathrm{d}t\\ &=\lim_{n \to \infty}\sum_{r=1}^{n}\dfrac{1}{nx+r}\\ &=\lim_{n \to \infty}\dfrac{1}{n}\sum_{r=1}^{n}\dfrac{1}{x+r/n}\\ &=\int_{0}^{1} \dfrac{1}{x+y} \mathrm{d}y\\ &=\log(1+x)-\log(x)\\ &=\boxed{\log\left(1+\dfrac{1}{x}\right)}\end{align}\]
NOTE : \(H_p\) is the \(p^{\text{th}}\) harmonic number.
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Hi Pratik , what's with you changing your profile pic . Don't like Federer anymore ?
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I'd never stop liking Federer :) I changed it just like that :P
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:)
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Proof is quite simple (simple properties of digamma functions)
We being with the property :
\(\displaystyle \psi{(y+1)}=\dfrac{1}{y} + \psi{(y)} \)
Put \(y=nx+1\) to get :
\( \displaystyle \psi{(nx+2)}= \dfrac{1}{nx+1} + \psi{(nx+1)} \)
\(\Rightarrow \displaystyle \psi{(nx+3)}= \dfrac{1}{nx+2} + \psi{(nx+2)} \)
Continuing like this we finally have :
\( \displaystyle \psi{(nx+n+1)}= \sum _{ r=1 }^{ n }{ \frac { 1 }{ nx+r } } + \psi{(nx+1)} \)
Finally we have :
\(\displaystyle \sum _{ r=1 }^{ n }{ \frac { 1 }{ nx+r } } = \psi{(nx+n+1)} - \psi{(nx+1)} \)
Now I am using another property converting riemann sum into an integral :
\( \displaystyle \lim _{ n\rightarrow \infty }{ \frac { 1 }{ n } \sum _{ r=1 }^{ n }{ f\left(\frac { r }{ n } \right) } } = \int _{ 0 }^{ 1 }{ f(x)dx } \)
Using this property we have :
\( \displaystyle \lim _{ n\rightarrow \infty }{\frac { 1 }{ n } \sum _{ r=1 }^{ n }{ \frac { 1 }{ x+\frac { r }{ n } } }} =\int _{ 0 }^{ 1 }{ \frac { dt }{ x+t } } \)
Finally getting :
\( \displaystyle \lim _{ n\rightarrow \infty }{ \psi (nx+n+1)-\psi (nx+1) } =log\left(1+\dfrac { 1 }{ x } \right) \)
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I got to that one till the sum but was very confused on how to get that sum. Nice, that was new to me! I didn't know about that conversion until now. Thanks!
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Shame on me , I was working on it from a totally different perspective .
Short and Sweet solution @Ronak Agarwal
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Ok, done! I've added my proof also.
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It must be applicable to complex numbers as well since the property of digamma function I used in the proof holds for complex numbers as well( you can always add 1 to complex number as well and hence create the summation)
Also when I converted riemann sum(right) into an integral the evaluation of that integral is governed by fundamental theorom of calculus and since the anti-derivative I calculated holds for complex numbers as well hence I believe your result is justified for complex numbers as well.
But since I am not being rigourous hence I may not be sure( If I am missing something)
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Is someone trying this problem, or should I post the proof?
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I am trying this.
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Sorry , I couldn't solve it past my initial efforts despite your Hint . BTW Wait for some hours, I'll reshare this note again ,I guess not many people have seen this note .
Looking forward to your proof \(\ddot\smile\)
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ok, as you say. And thanks for resharing it.
Therefore, I guess, I'll post the proof within next 12 hours. \(:)\). Will that be fine?
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Your Wish . Actually the next time I'm on Brilliant will be at night, so I guess it's fine by me \(\ddot\smile\)
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Nice observation skills , Kishlaya Jaiswal :)
We know that \( \psi (z+1) = F(x)\)
So using it , \[\lim_{n \rightarrow \infty} \psi(nx+n+1)-\psi(nx+1) \\= F(nx + n) - F(nx) \\= \frac{d}{dx} ( ln((nx+n)!)) - \frac{d}{dx} (ln((nx)!)) = \frac{d}{dx} ln (\dfrac{(nx+n)!}{(nx)!}) \]
Where \( \psi(x)\) is the Digamma Function and \(F(x)\) is the logarithmic derivative of the Factorial function defined as \(\dfrac{d}{dx} ln(x!)\)
Any help on how to proceed next ?
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ok, here's another hint for you : try computing the sum of following series in two different ways.
\[\lim_{n \rightarrow \infty} \sum_{r=0}^n \frac{1}{nx+r}\]
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I too got till this one except I think it should start from \(r=1\). But I am still unable to solve the sum.
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Ok, I'll try using your hint . Thanks
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As usual , your methods are the best :) Thanks and btw I realized the error in my calculations but yours and Pratik's methods are the best . I had tried Pratik's method just before I used the one that I posted . That one seemed to be so close to the final result that I proceeded with it . Can you please check if a proof using it is possible ? Personally, with my limited knowledge I don't think it's possible , but I just can't help stating that it looks quite similar to the final answer .
Thanks for the same :)
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