Matematika 2 Pegi Ushtrime Te Zgjidhura 💫

Përdorim kriterin e raportit: ( a_n = \frac{n!}{n^n} ) [ \frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!} = \frac{n+1}{(n+1)^{n+1}} \cdot n^n = \frac{n^n}{(n+1)^n} = \frac{1}{\left(1+\frac{1}{n}\right)^n} ] Kur ( n \to \infty ), ( \left(1+\frac{1}{n}\right)^n \to e ), pra ( \frac{a_{n+1}}{a_n} \to \frac{1}{e} \approx 0.368 < 1 ). Rrjedhimisht, seria konvergjon absolutisht.

( y(x) = \frac{1}{2} + C e^{-x^2} ) Ushtrimi 3: Seriali (kriteri i raportit) Studioni konvergjencën e serisë: [ \sum_{n=1}^{\infty} \frac{n!}{n^n} ] matematika 2 pegi ushtrime te zgjidhura

Konvergjon Ushtrimi 4: Integral i dyfishtë (koordinata polare) Llogaritni sipërfaqen e rrethit ( x^2 + y^2 \leq 4 ). Përdorim kriterin e raportit: ( a_n = \frac{n

( S = \iint_{x^2+y^2 \le 4} 1 , dA ) Në polare: ( x = r\cos\theta, y = r\sin\theta, \quad 0 \le r \le 2, \quad 0 \le \theta \le 2\pi ), Jakobi ( r ). [ S = \int_{0}^{2\pi} \int_{0}^{2} r , dr , d\theta = \int_{0}^{2\pi} \left[ \frac{r^2}{2} \right] {0}^{2} d\theta = \int {0}^{2\pi} 2 , d\theta = 4\pi ] ( S = \iint_{x^2+y^2 \le 4} 1 ,