\(B=\dfrac{1}{2}-\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{2}\right)^3-\left(\dfrac{1}{2}\right)^4+...-\dfrac{1}{2022}+\dfrac{1}{2023}\\ \Rightarrow B=\dfrac{2}{2^2}-\dfrac{1}{2^2}+\dfrac{2}{2^4}-\dfrac{1}{2^4}+...+\dfrac{2}{2^{2024}}-\dfrac{1}{2^{2024}}\)
\(\Rightarrow B=\dfrac{1}{2^2}+\dfrac{1}{2^4}+\dfrac{1}{2^6}+...+\dfrac{1}{2^{2024}}\)
\(\Rightarrow B=\dfrac{2^{2022}}{2^{2024}}+\dfrac{2^{2020}}{2^{2024}}+...+\dfrac{1}{2^{2024}}\\ \Rightarrow2^2B=\dfrac{2^{2024}}{2^{2024}}+\dfrac{2^{2022}}{2^{2024}}+...+\dfrac{2^2}{2^{2024}}\)
\(\Rightarrow4B-B=\dfrac{2}{2^{2024}}-\dfrac{1}{2^{2024}}\\ \Rightarrow3B=1-\left(\dfrac{2}{2^{2024}}+\dfrac{1}{2^{2024}}\right)\)
\(\Rightarrow3B=1-\dfrac{3}{2^{2024}}\\ \Rightarrow B=\dfrac{1-\dfrac{3}{2^{2024}}}{3}\)
\(\Rightarrow B=\dfrac{3\left(\dfrac{1}{3}-\dfrac{1}{2^{2024}}\right)}{3}\\ B=\dfrac{1}{3}-\dfrac{1}{2^{2024}}\)
