Question

E1. Sa se calculeze limita sirului
Punctul e

Answer 1

$an = \frac{ {1}^{4} + {2}^{4} + .... + {n}^{4} }{{n}^{5} } \\ an = \frac{1}{n} ( \frac{ {1}^{4} + {2}^{4} + .... + {n}^{4} }{ {n})^{4}} = \\ an = \frac{1}{n} \times ( { \frac{1}{n} )}^{4} \times.... \times ( { \frac{n}{n} })^{4}$

$an = ∑^{n} _{i = 1} ( \frac{i}{n})^{4} \times \frac{1}{n} \\ an = ∑^{n} _{i = 1} f(ci)(x_{i} - x_{i - 1}) \\ i = integrala \: s^{b} _{a}dx$

$f:[0,1] - > R \\ f(x) = {x}^{4}$

$dn = ( x_{0},x_{1},.....,x_{n}) \\ lim | |dn| | = 0, \\ oricare \: c \: apartine \: (x _{i - 1},x _{i}), \\ i = 1,n$

$an = ∑^{n} _{i = 1} f(ci)(x_{i} - x_{i - 1}) \\ i = integrala \: s^{b} _{a}dx$

$lim \: an = s ^{1} _{0}f(x)dx =s ^{1} _{0} {x}^{4} = \\ \frac{ {x}^{5} }{5}|^{1} _{0} = \frac{1}{5} - \frac{0}{5} = \frac{1}{5}$