Question

If someone can help with this it'd be nice, I want to be able to see from a geometric argument that ln1+ln2+ln3+….lnN is bounded below by the integral from of lnN from 1 to and N and bounded above by the integral of lnN from 1 to N+1. for all integers greater then or equal to 1. Thanks

Answer 1

well, consider a case of f(x) where F`(x)=f(x)

\(\large\color{slate}{\displaystyle\int\limits_{1}^{N}f(x)~dx}\) can be written as,

\(\large\color{slate}{\displaystyle\int\limits_{1}^{2}f(x)~dx+\int\limits_{2}^{3}f(x)~dx+\int\limits_{3}^{4}f(x)~dx+{\bf ....}+\int\limits_{n-1}^{N}f(x)~dx}\)

Answer 2

so you will take the
F(2)-F(1)+F(3)-F(2)+F(4)-F(3)+......+F(N)-F(N-1)
which is
-F(1) + F(N)
or
F(N)-F(1)

Answer 3

So you want a geometric interpretation of the following inequality:
\[\underbrace{\int_1^N \ln x\,dx}_A\le\underbrace{\sum_{k=1}^N\ln k}_B\le \underbrace{\int_1^{N+1}\ln x\,dx}_C\]