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I would like to know if there exists a closed form expression for the following series: $$\sum_{b=1}^{bmax} \sqrt{N-b^2}-b+1$$ where $ N>b^2$. I tried looking at the solutions to Sum of Square roots formula., but since in this case $N$ can be arbitrary, I am a bit stuck. Is it possible to find out and expression both the real and integer root case? Thanks in advance.

RTn
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  • Welcome to the site ! What looks to be interesting is the case where $b_{max}=\left\lfloor \sqrt{N}\right\rfloor$. Try with Excel : it looks to be very close to a straight line. If $b_{max}$ is anything, then ... ??? – Claude Leibovici Feb 20 '19 at 11:02
  • Thank you for your welcome. I will try in excel. But still there remains the case to have a closed form expression. – RTn Feb 20 '19 at 11:08
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    You changed the problem which now still more difficult. Concerning a closed form, do not dream too much. Cheers :-) – Claude Leibovici Feb 20 '19 at 11:12
  • Well, I am not having any high hopes. But, let us see! Yes, I changed the problem since I forgot to add that $b$ at the end. – RTn Feb 20 '19 at 11:24
  • Try the new problem using $b_{max}=\left\lfloor \sqrt{N}\right\rfloor$. Plot the function of $N$ and look at it from far away ! Cheers :-) – Claude Leibovici Feb 20 '19 at 11:27

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