Product series general formula?

Question

For a series like :

$$1^3 + 2^3 + 3^3 + 4^3 + \cdots + n^3$$

There is a general formula : $(n(n+1)/2)^2$

My question: Is there any general formula possible for following series :

$$1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4 \cdots n^n$$

Answer 1

let $$s_n=\prod_{r=1}^nr^r$$

Write$$s_n=1^1 \times2^2 \times 3^3 \times 4^4 \times\cdots \times n^n$$ Take $\log$ on both sides $$\ln (s_n)=\ln(1^1 \times2^2 \times 3^3 \times 4^4 \times\cdots \times n^n)$$

$$\ln (s_n)=\ln(1^1)+\ln(2^2)+\ln(3^3)+\ln(4^4)+\cdots +\ln(n^n)$$

$$\ln (s_n)=1\ln(1)+2\ln(2)+3\ln(3)+4\ln(4)+\cdots +n\ln(n)$$

And then See This Answer