Question

11,12 sau macar una dintre ele va rog dau coroana

Answer 1

11.

$\displaystyle S_1 = 1 -\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{5}-\dfrac{1}{6}+...+\dfrac{1}{19}-\dfrac{1}{20}\\ \\\\S_1 = \dfrac{1}{11}-\dfrac{1}{12}+\dfrac{1}{13}-\dfrac{1}{14}+...+\dfrac{1}{19}-\dfrac{1}{20}+1-\dfrac{1}{2}+\dfrac{1}{3}-...-\dfrac{1}{10}\\ \\ S_2 = \dfrac{1}{11}+\dfrac{1}{12}+\dfrac{1}{13}+...+\dfrac{1}{20}$

$\\S_2-S_1 = \dfrac{2}{12}+\dfrac{2}{14}+\dfrac{2}{16}+...+\dfrac{2}{20}-1+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{10} \\ \\ S_2 - S_1 = \dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+...+\dfrac{1}{10}-1+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{10}\\ \\ S_2-S_1 = \dfrac{2}{6}+\dfrac{2}{8}+\dfrac{2}{10}-1+\dfrac{1}{2}+\dfrac{1}{4}-\dfrac{1}{3}-\dfrac{1}{5} \\ \\ S_2-S_1 = \dfrac{1}{3}+\dfrac{1}{4}+\dfrac{1}{5}-\dfrac{1}{4}-\dfrac{1}{3}-\dfrac{1}{5}\\ \\ S_2-S_1 = 0 \\ \\\\\Rightarrow \boxed{S_1 = S_2}$