Approximating sum by integral
Integrate[(x^2-p^2)^1/2,{x,p + 1,(p^2+1)/2}]
This gives after simplification
1/8 (-1 + p^4 - 4 Sqrt[1 + 2 p] - 4 p Sqrt[1 + 2 p] +
4 p^2 (-2 Log[p] + Log[1 + p + Sqrt[1 + 2 p]]))
Better approximations can be obtained using Euler-Maclauren sum formula.
