( 1 + 1 1 ) ( 1 − 1 2 ) ( 1 + 1 3 ) ∗ ⋯ ∗ ( 1 + ( − 1 ) n + 1 n ) = 1 + ( − 1 ) n + 1 + 1 2 n {\displaystyle (1+{\frac {1}{1}})(1-{\frac {1}{2}})(1+{\frac {1}{3}})*\cdots *(1+{\frac {(-1)^{n+1}}{n}})=1+{\frac {(-1)^{n+1}+1}{2n}}}
L : ( 1 + ( − 1 ) n + 1 n ) = 1 + ( − 1 ) 2 1 = 2 R : 1 + ( − 1 ) n + 1 + 1 2 n = 1 + ( − 1 ) 2 + 1 2 = 1 + 1 = 2 2 = 2 {\displaystyle {\begin{aligned}&L:(1+{\frac {(-1)^{n+1}}{n}})=1+{\frac {(-1)^{2}}{1}}=2\\&R:1+{\frac {(-1)^{n+1}+1}{2n}}=1+{\frac {(-1)^{2}+1}{2}}=1+1=2\\&2=2\\\end{aligned}}}
( 1 + 1 1 ) ( 1 − 1 2 ) ( 1 + 1 3 ) ∗ ⋯ ∗ ( 1 + ( − 1 ) k + 1 k ) = 1 + ( − 1 ) k + 1 + 1 2 k {\displaystyle (1+{\frac {1}{1}})(1-{\frac {1}{2}})(1+{\frac {1}{3}})*\cdots *(1+{\frac {(-1)^{k+1}}{k}})=1+{\frac {(-1)^{k+1}+1}{2k}}}
( 1 + 1 1 ) ( 1 − 1 2 ) ( 1 + 1 3 ) ∗ ⋯ ∗ ( 1 + ( − 1 ) k + 1 k ) ⏟ = 1 + ( − 1 ) k + 1 + 1 2 k ∗ ( 1 + ( − 1 ) k + 2 k + 1 ) = 1 + ( − 1 ) k + 2 + 1 2 ( k + 1 ) [ 1 + ( − 1 ) k + 1 + 1 2 k ] ∗ [ 1 + ( − 1 ) k + 2 k + 1 ] = 1 + ( − 1 ) k + 2 + 1 2 ( k + 1 ) 1 + ( − 1 ) k + 2 k + 1 + ( − 1 ) k + 1 + 1 2 k + [ ( − 1 ) k + 1 + 1 ] [ ( − 1 ) k + 2 ] 2 k ∗ ( k + 1 ) = 1 + ( − 1 ) k + 2 + 1 2 ( k + 1 ) 2 k ( − 1 ) k + 2 + ( k + 1 ) ( − 1 ) k + 1 + k + 1 + [ ( − 1 ) k + 1 + 1 ] [ ( − 1 ) k + 2 ] = k ( − 1 ) k + 2 + k 2 k ( − 1 ) k + 2 + k ∗ ( − 1 ) k + 1 + ( − 1 ) k + 1 + k + 1 + ( − 1 ) 2 k + 3 + ( − 1 ) k + 2 = k ( − 1 ) k + 2 + k 2 k ( − 1 ) k + 2 − k ∗ ( − 1 ) k + 2 ⏟ k ( − 1 ) k + 2 − ( − 1 ) k + 2 + 1 + ( − 1 ) 2 k ∗ ( − 1 ) 3 + ( − 1 ) k + 2 = k ( − 1 ) k + 2 k ( − 1 ) k + 2 + 1 − ( − 1 ) 2 k ⏟ ( − 1 ) 2 k = 1 = k ( − 1 ) k + 2 1 − 1 = 0 0 = 0 {\displaystyle {\begin{aligned}&\underbrace {(1+{\frac {1}{1}})(1-{\frac {1}{2}})(1+{\frac {1}{3}})*\cdots *(1+{\frac {(-1)^{k+1}}{k}})} _{=1+{\frac {(-1)^{k+1}+1}{2k}}}*(1+{\frac {(-1)^{k+2}}{k+1}})=1+{\frac {(-1)^{k+2}+1}{2(k+1)}}\\&[1+{\frac {(-1)^{k+1}+1}{2k}}]*[1+{\frac {(-1)^{k+2}}{k+1}}]=1+{\frac {(-1)^{k+2}+1}{2(k+1)}}\\&1+{\frac {(-1)^{k+2}}{k+1}}+{\frac {(-1)^{k+1}+1}{2k}}+{\frac {[(-1)^{k+1}+1][(-1)^{k+2}]}{2k*(k+1)}}=1+{\frac {(-1)^{k+2}+1}{2(k+1)}}\\&2k(-1)^{k+2}+(k+1)(-1)^{k+1}+k+1+[(-1)^{k+1}+1][(-1)^{k+2}]=k(-1)^{k+2}+k\\&2k(-1)^{k+2}+k*(-1)^{k+1}+(-1)^{k+1}+k+1+(-1)^{2k+3}+(-1)^{k+2}=k(-1)^{k+2}+k\\&\underbrace {2k(-1)^{k+2}-k*(-1)^{k+2}} _{k(-1)^{k+2}}{\color {blue}-(-1)^{k+2}}+1+(-1)^{2k}*(-1)^{3}{\color {blue}+(-1)^{k+2}}=k(-1)^{k+2}\\&k(-1)^{k+2}+1-\underbrace {(-1)^{2k}} _{(-1)^{{\color {red}2}k}=1}=k(-1)^{k+2}\\&1-1=0\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
![{\displaystyle {\begin{aligned}&\underbrace {(1+{\frac {1}{1}})(1-{\frac {1}{2}})(1+{\frac {1}{3}})*\cdots *(1+{\frac {(-1)^{k+1}}{k}})} _{=1+{\frac {(-1)^{k+1}+1}{2k}}}*(1+{\frac {(-1)^{k+2}}{k+1}})=1+{\frac {(-1)^{k+2}+1}{2(k+1)}}\\&[1+{\frac {(-1)^{k+1}+1}{2k}}]*[1+{\frac {(-1)^{k+2}}{k+1}}]=1+{\frac {(-1)^{k+2}+1}{2(k+1)}}\\&1+{\frac {(-1)^{k+2}}{k+1}}+{\frac {(-1)^{k+1}+1}{2k}}+{\frac {[(-1)^{k+1}+1][(-1)^{k+2}]}{2k*(k+1)}}=1+{\frac {(-1)^{k+2}+1}{2(k+1)}}\\&2k(-1)^{k+2}+(k+1)(-1)^{k+1}+k+1+[(-1)^{k+1}+1][(-1)^{k+2}]=k(-1)^{k+2}+k\\&2k(-1)^{k+2}+k*(-1)^{k+1}+(-1)^{k+1}+k+1+(-1)^{2k+3}+(-1)^{k+2}=k(-1)^{k+2}+k\\&\underbrace {2k(-1)^{k+2}-k*(-1)^{k+2}} _{k(-1)^{k+2}}{\color {blue}-(-1)^{k+2}}+1+(-1)^{2k}*(-1)^{3}{\color {blue}+(-1)^{k+2}}=k(-1)^{k+2}\\&k(-1)^{k+2}+1-\underbrace {(-1)^{2k}} _{(-1)^{{\color {red}2}k}=1}=k(-1)^{k+2}\\&1-1=0\\&0=0\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4702c3129d0689253cf59a1e2cb4b9358706188)