Question

$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{n}, n\geq 1$

Imi poate spune cineva va rog de ce sirul asta este crescator?

Answer 1

[tex]\text{Daca notam sirul cu }a_n \text{ avem ca }\\ a_{n+1}-a_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots+\dfrac{1}{n+1}-1-\dfrac{1}{2}-\dfrac{1}{3}-\ldots -\dfrac{1}{n}=\dfrac{1}{n+1}\\ \text{Evident }\dfrac{1}{n+1}\ \textgreater \ 0\text{ ceea ce implica }a_{n+1}-a_{n}\ \textgreater \ 0,\text{deci }a_n\text{ este} \\ \text{crescator.} [/tex]