Is there a general formula to calculate the sum of the squares logarithms of first n natural numbers?
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Yaswanth Sai
on 14 Apr 2018
Answered: Walter Roberson
on 14 Apr 2018
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
on 14 Apr 2018
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
on 14 Apr 2018
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
on 14 Apr 2018
Hmm, I do not know a general formula for that, Google will be your best friend in this case.
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