Textbook integration problem. Unclear how the limit was arrived at.

Question

I am looking at this text:

and I'm not clear how (b) was arrived at. (a) makes sense to me. How did they go from:

$$\sum_{i=1}^n e^{1 + 2i/n}$$

to the long fraction on the right? What is even going on?

Answer 1

If you wanted to get that expression by hand, you could note that the sum is geometric and use (with some modifications) the general rule

$$\sum_{i=0}^m ar^i = a\frac{1-r^{m+1}}{1-r}\qquad\textrm{when $|r| < 1$.}$$

Answer 2

$\frac {2}{n}\sum_\limits{i=1}^n e^{1+\frac {2i}{n}} = $$\frac {2}{n}(e^{1+\frac {2}{n}} + e^{1+\frac {4}{n}} + e^{1+\frac {6}{n}} +\cdots + e^{3})\\ \frac {2}{n}(e)(e^{\frac 2n} + e^{\frac {4}{n}} + e^{\frac {6}{n}} +\cdots + e^{2})$

And the factor on the right is a geometric series.

i.e. $\sum_\limits{i=1}^{n} r^{i} = \frac {r(1-r^n)}{1-r}$ and let $r = e^\frac{2}{n}$

$\frac {2}{n}(e)\frac {e^{\frac 2n}(1 - e^{2})}{1-e^\frac {2}{n}} = \frac {2}{n}\frac {e^{\frac {3n+ 2}{n}} - e^{\frac {n + 2}{n}}}{e^\frac {2}{n} - 1}$