Δ x = x 1 − x 0 {\displaystyle \Delta x=x_{1}-x_{0}}
∑ i = 1 ∞ 1 2 i {\displaystyle \sum _{i=1}^{\infty }{\frac {1}{2^{i}}}}
= 1 2 + 1 4 + 1 8 + … {\displaystyle ={\frac {1}{2}}+{\frac {1}{4}}+{\frac {1}{8}}+\ldots }
= 1 {\displaystyle =1}
∑ i = 0 ∞ ( − 1 2 ) i {\displaystyle \sum _{i=0}^{\infty }\left(-{\frac {1}{2}}\right)^{i}}
= 1 − 1 2 + 1 4 − 1 8 + 1 16 − 1 32 + … {\displaystyle =1-{\frac {1}{2}}+{\frac {1}{4}}-{\frac {1}{8}}+{\frac {1}{16}}-{\frac {1}{32}}+\ldots }
= ( 1 + 1 4 + 1 16 + … ) − ( 1 2 + 1 8 + 1 32 + … ) {\displaystyle =\left(1+{\frac {1}{4}}+{\frac {1}{16}}+\ldots \right)-\left({\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}+\ldots \right)}
= [ 1 + 1 2 + 1 4 + … − ( 1 2 + 1 8 + … ) ] − ( 1 2 + 1 8 + 1 32 + … ) {\displaystyle =\left[1+{\frac {1}{2}}+{\frac {1}{4}}+\ldots -\left({\frac {1}{2}}+{\frac {1}{8}}+\ldots \right)\right]-\left({\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}+\ldots \right)}
= 2 − 2 ( 1 2 + 1 8 + 1 32 + … ) {\displaystyle =2-2\left({\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}+\ldots \right)}
= 2 − ( 1 + 1 4 + 1 16 + … ) {\displaystyle =2-\left(1+{\frac {1}{4}}+{\frac {1}{16}}+\ldots \right)}
= 2 − ∑ i = 0 ∞ ( 1 4 ) i {\displaystyle =2-\sum _{i=0}^{\infty }\left({\frac {1}{4}}\right)^{i}}
= 2 − 1 1 − 1 4 {\displaystyle =2-{\frac {1}{1-{\frac {1}{4}}}}}
= 2 − 1 3 4 {\displaystyle =2-{\frac {1}{\frac {3}{4}}}}
= 2 − 4 3 = 2 3 {\displaystyle =2-{\frac {4}{3}}={\frac {2}{3}}}
![{\displaystyle =\left[1+{\frac {1}{2}}+{\frac {1}{4}}+\ldots -\left({\frac {1}{2}}+{\frac {1}{8}}+\ldots \right)\right]-\left({\frac {1}{2}}+{\frac {1}{8}}+{\frac {1}{32}}+\ldots \right)}](/media/api/rest_v1/media/math/render/svg/cf9cf5078e3f6f651a489a5dcef1ee5854c5e2b4)
