Zadanie 1
Iloraz ciągu geometrycznego:
[tex]q=\dfrac{3x}{1}=3x[/tex]
Szereg jest zbieżny, gdy:
[tex]|q|<1[/tex]
Stąd:
[tex]|3x|<1\\\\3x<1\quad\text{i}\quad3x>-1\\\\x<\dfrac{1}{3}\quad\text{i}\quad x>-\dfrac{1}{3}\\\\\boxed{x\in\Big(-\dfrac{1}{3},\dfrac{1}{3}\Big)}[/tex]
Zadanie 2
Iloraz ciągu geometrycznego:
[tex]q=\dfrac{\Big(x-\dfrac{1}{8}\Big)^2}{1}={\Big(x-\dfrac{1}{8}\Big)^2[/tex]
Wobec tego:
[tex]\Big|{\Big(x-\dfrac{1}{8}\Big)^2\Big|<1[/tex]
[tex]{\Big(x-\dfrac{1}{8}\Big)^2<1[/tex]
[tex]{\Big(x-\dfrac{1}{8}\Big)^2-1^2<0[/tex]
[tex]{\Big(x-\dfrac{1}{8}+1\Big){\Big(x-\dfrac{1}{8}-1\Big)<0[/tex]
[tex]\Big(x+\dfrac{7}{8}\Big)\Big(x-\dfrac{9}{8}\Big)<0\\\\\\x_1=-\dfrac{7}{8},\quad x_2=\dfrac{9}{8}\\\\\\\boxed{x\in\Big(-\dfrac{7}{8},\dfrac{9}{8}\Big)}[/tex]
