Sum of Square roots formula.

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.

Answer

The definition of harmonic numbers is H(a)p=pi=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)+124p+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

Attribution
Source : Link , Question Author : lk42392 , Answer Author : Claude Leibovici

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