Question

DAU COROANĂ!
Subpunctele e), f) și g) din poză. Mulțumesc anticipat.

Answer 1

e)

$\displaystyle a_{n+1}-a_n=\frac11+...+\frac1n+\frac1{n+1}-\left(\frac11+...+\frac1n\right)=\frac1{n+1}>0,\forall n\in\mathbb{N}\Rightarrow \left(a_n\right)_{n\in\mathbb{N}}\text{cresc\u ator}$

f)

$\displaystyle\frac{a_{n+1}}{a_n}=\frac{\displaystyle\left(1-\frac{1}{2}\right)\cdot...\cdot\left(1-\frac1{2^n}\right)\cdot\left(1-\frac{1}{2^{n+1}}\right)}{\displaystyle\left(1-\frac{1}{2}\right)\cdot...\cdot\left(1-\frac1{2^n}\right)}=1-\frac{1}{2^{n+1}}<1,\forall n\in\mathbb{N}\Rightarrow\left(a_n\right)_{n\in\mathbb{N}}\text{ descresc\u ator}$

g)

$\displaystyle\frac{a_{n+1}}{a_n}=\frac{\displaystyle\left(1-\frac1{2^2}\right)\cdot...\cdot\left[1-\frac1{(n+1)^2}\right]\cdot\left[1-\frac1{(n+2)^2}\right]}{\displaystyle\left(1-\frac1{2^2}\right)\cdot...\cdot\left[1-\frac1{(n+1)^2}\right]}=1-\frac{1}{(n+2)^2}<1,\forall n\in\mathbb{N}\Rightarrow\left(a_n\right)_{n\in\mathbb{N}}\text{descresc\u ator}$