# 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 $$sum r ^ beta leq sum r ^ beta ln r leq sum r ^ { beta + 1}$$?