I have this problem - Express limit as a definite integral. The limit is: $$\lim_{n\to \infty}\sum_{i=0}^n\frac{3+\ln(n+i)-\ln(n)}{n}$$
I can't figure out how to make a definite integral out of it. So far I have: $$\frac{1}{n} =dx$$ $$3+\ln(n+i)-\ln(n) = a+ \frac{b-a}{n}\cdot i $$ (where b & a are limits of integration). I know that $\ln(x) - \ln(y) = \ln(\frac{x}{y})$, so I suppose it should go like $$3+\ln(\frac{n+i}{n})$$ but then I just land with $\infty$. Help?
With $\int\limits_0^x f(t)dt=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=1}^n f(x\frac{i}{n})$ you can solve your problem.