# How do you find a formula for the sum n terms Sigma(2i)/n(2/n) and then find the limit as n->oo?

Nov 20, 2016

${\sum}_{i = 1}^{n} \frac{2 i}{n} \left(\frac{2}{n}\right) = 2 + \frac{2}{n}$

${\lim}_{n \rightarrow \infty} {\sum}_{i = 1}^{n} \frac{2 i}{n} \left(\frac{2}{n}\right) = 2$

#### Explanation:

We will need the standard results:
${\sum}_{r = 1}^{n} r = \frac{1}{2} n \left(n + 1\right)$

Let

${S}_{n} = {\sum}_{i = 1}^{n} \frac{2 i}{n} \left(\frac{2}{n}\right)$
$\therefore {S}_{n} = \frac{4}{n} ^ 2 {\sum}_{i = 1}^{n} i$
$\therefore {S}_{n} = \frac{4}{n} ^ 2 \left\{\frac{1}{2} n \left(n + 1\right)\right\}$
$\therefore {S}_{n} = \frac{2}{n} \left(n + 1\right)$
$\therefore {S}_{n} = 2 + \frac{2}{n}$

Hence, ${\sum}_{i = 1}^{n} \frac{2 i}{n} \left(\frac{2}{n}\right) = 2 + \frac{2}{n}$

Consequently,

${\lim}_{n \rightarrow \infty} {S}_{n} = {\lim}_{n \rightarrow \infty} \left(2 + \frac{2}{n}\right)$
${\lim}_{n \rightarrow \infty} {S}_{n} = {\lim}_{n \rightarrow \infty} \left(2\right) + {\lim}_{n \rightarrow \infty} \left(\frac{2}{n}\right)$
${\lim}_{n \rightarrow \infty} {S}_{n} = 2 + 0$
${\lim}_{n \rightarrow \infty} {S}_{n} = 2$