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## Homework Statement

Suppose f is continuously differentiable, and a < b. The sequence is defined as follows:

[itex]a_{n} = n\int_a^b \! f(x) \, \mathrm{d} x - n(\frac{b-a}{n})\displaystyle\sum\limits_{i=1}^n f(a + \frac{b-a}{n}i)[/itex]

## The Attempt at a Solution

I've been busting my *** with this one for well over 3 hours now, and thus far the only thing I was able to is to use the theorem that states that for some y, [itex]f(y)(b-a)=\int_a^b \! f(x) \, \mathrm{d} x[/itex].

I have then expanded the sequence, took apart the summation to group the same values together, and ultimately got:

[itex]a_{n} = (b - a)((nf(y) - \displaystyle\sum\limits_{i=1}^n f(a + \frac{b-a}{n}i)) = ... = (b - a)((f(y) - f(a + \frac{b-a}{n}) + ... + (f(y) - f(b)).[/itex]

I'm unable to go anywhere from here I'm also not sure whether I should've shown that the sequence converges first, and then use a different approach to find its limit. I've been trying to show it converges, but I don't think it's monotone (even though it's bounded), and I also can't show that it's Cauchy. I figured the approach I took would take care of both the fact that it converges and the limit, but so far I'm not getting any of them.

Thanks in advance for any help!