5
$\begingroup$

If $f(x)$ is continuous in $[a,b]$, prove that $ \displaystyle \lim_{n \to \infty} \dfrac{b-a}{n} \displaystyle \sum^n _{k=1} f\left( a + \dfrac{k(b-a)}{n} \right) = \displaystyle \int_a ^ b f(x)dx$

This is my first time I'm exposed to these type of math problems (being a high schooler), so I don't really know how to tackle this. Can anyone point me in the right direction?

What I tried:

I just tried to see the logic behind the RHS. I see that $\dfrac{b-a}{n}$ divides the interval a,b into n rectangles. What I totally don't understand is why the height of these rectangles is given by the summation of $ f\left(a + \dfrac{k (b-a)}{n}\right)$

$\endgroup$
0

1 Answer 1

Reset to default
2
$\begingroup$

You will need to use the fact that $f$ is uniformly continuous on $[a,b]$.

$\endgroup$
2
  • $\begingroup$ That is why I offered this minimal hint. I didn't want to spoil it for you. $\endgroup$
    – ncmathsadist
    Feb 26, 2013 at 1:09
  • $\begingroup$ Have fun with this stuff. It is totally cool! $\endgroup$
    – ncmathsadist
    Feb 26, 2013 at 1:09

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .