# Simple Stirling:

1. Oct 4, 2007

### Monochrome

1. The problem statement, all variables and given/known data

$$f(1)+f(2)...+f(n-1) =< \int_{0}^{n} f(x) dx =< f(2) +f(3)+...+f(n)$$

is increasing and continuous on [1, inf)
I'm meat to prove the above, the idea I had was to use the trapezium rule to get an approximation of the integral, but since f''(x) can be either negative or positive I'm stuck as to how to do that. Also I'm in a first year course and I haven't yet learned about Bernoulli numbers or the like, which is what came up when I was looking around for this problem.

Edit: Would using left and right Riemann sums solve this? I can get the inequality but does increasing in this case mean non-decreasing?

Last edited: Oct 4, 2007
2. Oct 4, 2007

### NateTG

Are you sure that the integral isn't from 1 to n?

I'd be inclined to start by trying to show that if $f(x)$ is increasing, then:
$$1 \times f(n) \leq \integral_{n}{n+1} f(n) \leq 1 \times f(n+1)$$

The sums do correspond to left and right Riemann sums, but that's not going to be part of the proof unless you have a specific theorem about Riemann sums and increasing functions available.