Sum of the square root of integers from 1 to n

AI Thread Summary
The discussion focuses on finding a formula to calculate the sum of the square roots of integers from 1 to n. The original poster found a complex identity involving Bernoulli numbers on Wikipedia but found it too complicated to understand. They attempted to derive a simpler formula using Excel, resulting in a power regression of 0.701n^(1.492), which is somewhat accurate but not perfect. A suggestion was made to refer to MathWorld's article on power sums for a more general approach, as Faulhaber's formula is specifically for integer powers. Overall, the conversation highlights the need for a more accessible method to calculate the desired sum.
Thirit
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Homework Statement


I want to know what's the formula to calculate the sum of the square root of integers from 1 to n.
I got an identity from wikipedia but its too complicated for me, it involves bernoulli's number, i don't know what is that.


Homework Equations


05228b0f23694df78a466dd7007152ca.png



The Attempt at a Solution


In excel i managed to get a power regression and i got the formula 0.701n^(1.492), its kind of accurate but not 100%.

I hope someone could help me.
Thanks
 
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Bernoulli numbers

The Wikipedia page entitled "Bernoulli number" has the definition:
Bernoulli numbers may be calculated by using the following recursive formula:
\sum_{j=0}^m\left(\begin{array}{ c }<br /> m+1 \\<br /> j<br /> \end{array}\right)B_j=0
for m > 0, and B0 = 1.
 
Thirit said:
[I want to know what's the formula to calculate the sum of the square root of integers from 1 to n.
I got an identity from wikipedia but its too complicated for me, it involves bernoulli's number, i don't know what is that.
Exactly what Bernoulli numbers are (but see EnumaElish's post) is a bit irrelevant here because that identity, known as Faulhaber's formula, is only valid for integer powers.

What you want is something more general. See the mathworld article on power sums, http://mathworld.wolfram.com/PowerSum.html" , particularly equations 10 through 12.
 
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