I would like to know if there is formula to calculate sum of series of square roots √1+√2+⋯+√n like the one for the series 1+2+…+n=n(n+1)2.
Thanks in advance.
The definition of harmonic numbers is H(−a)p=p∑i=1ia When a is not a positive integer, there is no closed form but, as Yves Daoust commented, there are quite nice expansions.
For example, if n=12 as in the post, you have H(−12)p=2p3/23+√p2+ζ(−12)+124√p+O((1p)5/2) where ζ(−12)≈−0.2078862250.
For example, for p=10, the exact value is ≈22.46827819 while the above approximation gives ≈22.46827983. By itself, the first term already gives 21.0819; the sum of first and second term gives ≈22.6629. For p=100, the approximation leads to 12 exact significant figures.
There are similar expansions for any value of the exponent a