Help! Converting a Riemann Sum to a Definite Integral

Question

Can someone please explain how to convert this into a definite integral in the form

$$\lim _{ n\rightarrow \infty }{ \left\{ \ln { \sqrt [ n ]{ \left( n+1 \right) \left( n+2 \right) \left( n+3 \right) ...\left( 2n \right) } -n\ln { \sqrt [ n ]{ n } } } \right\} } $$

Related ; http://math.stackexchange.com/questions/845415/find-lim-n-rightarrow-infty-fracnn1-n — lab bhattacharjee, Jun 24 '14 at 05:42

score 4 · Accepted Answer · answered Jun 24 '14 at 05:32

4

This is nothing but $$\lim_{n \to \infty} \frac{1}{n}\{\log\frac{n+1}{n}+\log\frac{n+2}{n}+...+\log\frac{2n}{n}\}=\lim_{n \to \infty}\frac{1}{n}\sum_{k=1}^n\log \frac{n+k}{n}=\int_0^1 \log (1+x)dx$$

answered Jun 24 '14 at 05:32

tattwamasi amrutam

13,040

Thank you for providing the requested information. – ZellAllon Jun 24 '14 at 05:45

Help! Converting a Riemann Sum to a Definite Integral

1 Answers1