Is there a general formula to calculate the sum of the squares logarithms of first n natural numbers?
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Is there a general formula for the following sequence?
S(n) = [log(1)]^2 + [log(2)]^2 + ......... + [log(n-1)]^2 + [log(n)]^2
and similarly for the sum of cubes of logarithms of first n natural numbers and if there is one please let me know the procedure you have taken to arrive at the solution so that I can extend that to higher orders.
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Walter Roberson
2018-4-14
No there is no formulas for that. There are some inequalities known and there are some conjectures. One approximation is discussed in the link
https://www.physicsforums.com/threads/sum-of-log-squared-terms.627139/
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Birdman
2018-4-14
Symbolic Toolbox and its features are best for you to get what you want. For instance:
syms S(n,x) m
S(n,x)=symsum(log(n).^x,n,1,n)
This code simply defines symbolic variables m,n,x and symbolic function S which is a function of n and x. Then, we define a series which sums log(n)^x starting from 1 to n and also lets you define the power of logx depending on your input. A numeric example:
>> S(5,3)
ans =
log(2)^3 + log(3)^3 + log(4)^3 + log(5)^3
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Birdman
2018-4-14
Hmm, I do not know a general formula for that, Google will be your best friend in this case.
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