I have a question here in which I need to explicitly write down a sequence $f_n(x)$ that can approximate $e^x$.
From reading, I known that I need to pick a partition sequence of $x_k$ so that I can take $x_k^*\,\epsilon\,[x_k,x_{k+1})$ so that I can do $f_n(x) = f(x_k^*)$.
The bit I'm having trouble with is how exactly do I know what $f_n(x)$ to pick, and to 'write down explicitly' do I have to pick a maximum n, and partition $x_k$'s?
Any help would be greatly appreciated.
Edit: I've let $x_k = a + \frac{k(b-a)}{n} $ for all k = 0, 1,..., n to split into even partitions. I then defined the step functions to be $$f_n(x) = f(x_k^*)$$ where $x_k^* = \frac{x_k+x_{k+1}}{2}$. From this then: $$f_n(x) = e^{x_k^*} = e^\frac{x_k+x_{k+1}}{2}$$ on each interval [$x_k,x_{k+1}$). Using this we get $\int_a^b{e^x dx} = \sum_{k=0}^n{f(x_k^*)\frac{b-a}{n}} = \frac{b-a}{n}\sum_{k=0}^n{e^\frac{x_k+x_k+1}{2}}$.
Past this point I don't know how to approximate the integral.