Riemann sum for $\int_0^a e^x dx$.

Question

$$\int_0^a e^x dx$$

With the help of a previous example ($x^k$) I managed to solve I came this far:

Partition of $[0,a[$ into $n$ equilateral subintervals $[x_k,x_{k+1}[~: x_k=\frac{ka}{n}$ with $0 \leq k < n$. Then the Riemann sum is equal to $$\frac{a}{n}\sum_{k=0}^{n-1} e^{ka/n}$$

Using the sum for the geomatric series: $$\frac{a}{n}\cdot\frac{1-e^{a}}{1-e^{a/n}}$$

However, here I'm stuck—differentiating only makes it messier (L'Hôpital). What should I do/how to show that $\frac{a}{n(1-e^{a/n})}$ tends to $-1$?

Answer 1

HINT:

$e^{a/n}=1+\frac an+O\left(\frac an\right)^2$

Answer 2

Set $\dfrac an=h$

$$\lim_{n\to\infty}n\left(1-e^{a/n}\right)=-a\cdot\lim_{h\to0}\dfrac{e^h-1}h=?$$