sequence and series – List of asymptotic expressions for $ sum_r r ^ beta ln r $?

Expression for $ beta = $ 1 in $ sum_r r ^ beta ln r $

I recently had the following idea to use the identity below:

$$ (1! 2! 3! Points n!) (12 ^ 2 3 ^ 3 4 ^ 4 points n ^ n) = n! ^ N $$

Divide both sides by $ n ^ {n (n + 1) / 2} $ and using $ 1 + 2 + 3 points + n = frac {n (n + 1)} {2} $:

$$ (1! 2! 3! Dots n!) ( Frac {1} {n} ( frac {2} {n}) ^ 2 ( frac {3} {n}) ^ 3 ( frac {4} {n}) ^ 4 dots ( frac {n} {n}) ^ n) = frac {n! ^ N} {n ^ {n (n + 1) / 2}} $$

Raise both sides to power $ 1 / n:

$$ (1! 2! 3! Dots n!) ^ { Frac {1} {n}} (( frac {1} {n}) ^ { frac {1} {n}} ( frac {2} {n}) ^ { frac {2} {n}} ( frac {3} {n}) ^ { frac {3} {n}} ( frac {4} {n}) ^ { frac {4} {n}} dots ( frac {n} {n}) ^ { frac {1} {n}}) = frac {n!} {n ^ {(n + 1) / 2}} $$

taking $ ln $ on both sides:

$$ sum_ {r = 1} ^ n ln (r!) frac {1} {n} + sum_ {r = 1} ^ n frac {r} {n} ln ( frac {r } {n}) = ln ( frac {n!} {n ^ {(n + 1) / 2}}) $$

Divide both sides with $ 1 / n:

$$ sum_ {r = 1} ^ n ln (r!) frac {1} {n ^ 2} + sum_ {r = 1} ^ n frac {r} {n} ln ( frac {r} {n}) frac {1} {n} = frac {1} {n} ln ( frac {n!} {n ^ {(n + 1) / 2}}) $$

In the limit $ n to infty $ then $ sum_ {r = 1} ^ n frac {r} {n} ln ( frac {r} {n}) frac {1} {n} to int_0 ^ 1 x ln x dx = -1/4 $. Therefore,

$$ sum_ {r = 1} ^ n ln (r!) frac {1} {n ^ 2} – frac {1} {4} sim frac {1} {n} ln ( frac {n!} {n ^ {(n + 1) / 2}}) $$

So,

$$ sum_ {r = 1} ^ n ln (r!) frac {1} {n ^ 2} sim ln ( frac {n!} {n ^ {(n + 1) / 2} }) + frac {1} {4} $$

Using the Stirling approximation on the R.H.S:

$$ sum_ {r = 1} ^ n ln (r!) frac {1} {n ^ 2} sim frac {n-1} {2} ln n + frac {1} {4 } $$

Using Stirling's approximation on the L.H.S:

$$ sum_ {r = 1} ^ n (r r + r + O ( ln r)) frac {1} {n ^ 2} sim frac {n-1} {2} ln n + frac {1} {4} $$

Simplify both sides:

$$ sum_ {r = 1} ^ nr ln r sim n ^ 2 ( frac {n-1} {2}) ln n – frac {n (n-2)} {4} + O ( nn!) $$

Question

I did not consider the error when converting to integral when I did the step "In the limit $ n to infty $ then $ sum_ {r = 1} ^ n frac {r} {n} ln ( frac {r} {n}) frac {1} {n} to int_0 ^ 1 x ln x dx = -1/4 $".. What is the error? Is there a list of asymptotic expressions for $ sum r ^ beta ln r $ without using $ sum r ^ beta leq sum r ^ beta ln r leq sum r ^ { beta + 1}?