Question

Por favor poderiam me ajudar

Seja a soma dada pela expressão.


         n
s(n)=∑(3i+3)
       i=1 n²
lim s/(n)
0n→∞ vale

Answer 1

pelo oq eu entendi é :

$s(n)=\sum_{i=1}^{n} \left( \frac{3i+3}{n^2} \right)\\\\s(n)= \frac{3}{n^2} \sum_{i=1}^{n} \left(i+1 \right) \\\\ \boxed{\boxed{s(n)= \frac{3}{n^2} \left[\sum_{i=1}^{n}(i) + \sum_{i=1}^{n}(1)\right ] }}$

resolvendo

$\sum_{i=1}^{n}(i) = \frac{(1+n)*n}{2} \\\\r=1\\a_1=1\\a_n = n\\\\ \boxed{S_n= \frac{(a_1+a_n)*n}{2} = \frac{(1+n)*n}{2} }$

temos
$s(n)= \frac{3}{n^2} \left( \frac{(1+n)*n}{2}+n \right)\\\\s(n)= \frac{3}{n}\left( \frac{1+n}{2}+1 \right) \\\\\boxed{\boxed{s(n)= \frac{3n+9}{2n} }}$

calculando o limite
$\lim_{n \to \infty} \frac{3n+9}{2n} \\\\ = \lim_{n \to \infty} \frac{n*(3+ \frac{9}{n} )}{2n} \\\\ \lim_{n \to \infty} \frac{3+ \frac{9}{n} }{2}= \frac{3+0}{2}= \frac{3}{2}$