How to solve $\lim_{x\rightarrow \infty }\dfrac {2} {x}\sum _{k\rightarrow 1}^{x}\ln \left( \dfrac {x+k} {x}\right)$?

Question

Let k be positive integers $\lim_{x\rightarrow \infty }\dfrac {2} {x}\sum _{k\rightarrow 1}^{x}\ln \left( \dfrac {x+k} {x}\right)$

Answer 1

$$\lim_{x\rightarrow \infty }\dfrac {2} {x}\sum_{k\rightarrow 1}^{x}\ln \left( \dfrac {x+k} {x}\right)=\\ \lim_{x\rightarrow \infty }2\sum _{k\rightarrow 1}^{x}\frac {1}{x}\ln \left( \frac{x(1+\frac{k}{x})}{x}\right)$$ Now: by definition of integral $$2\lim_{x \to \infty} 2\sum_{1}^{x}\frac{1}{x}\ln(1+\frac{k}{x})\\=2\int_{0}^{1}ln(1+t)dt =\\$$using integration by part $$=2((1+t)ln(1+t)-\int(1+t)(ln(1+t))'dt)\\=2((1+t)ln(1+t)-\int(1+t)\frac{1}{1+t}dt)\\=2((1+t)ln(1+t)-t)$$