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In which step did we first make a mistake?
0=∑i=0∞0(1)=∑i=0∞(1−1)(2)=∑i=0∞1−∑i=0∞1(3)=(1+∑i=1∞1)−∑j=0∞1(4)=1+∑j=0∞1−∑j=0∞1(5)=1+∑j=0∞(1−1)(6)=1(7) \begin{array} { l l l } 0 & = \displaystyle \sum_{i=0}^\infty 0 & (1) \\ & = \displaystyle \sum_{ i=0}^ \infty (1 -1) & (2) \\ & =\displaystyle \sum_{ i=0}^ \infty 1 - \displaystyle \sum_{i=0}^\infty 1 & (3) \\ & = \left ( 1 + \displaystyle \sum_{ {\color{#D61F06}i=1}}^ \infty 1\right) - \displaystyle \sum_{j=0}^\infty 1 & (4) \\ & = 1 +\displaystyle \sum_{ j=0}^ \infty 1 - \displaystyle \sum_{j=0}^\infty 1 & (5) \\ & = 1 +\displaystyle \sum_{ j=0}^ \infty (1 - 1) & (6) \\ & = 1 & (7) \\ \end{array} 0=i=0∑∞0=i=0∑∞(1−1)=i=0∑∞1−i=0∑∞1=(1+i=1∑∞1)−j=0∑∞1=1+j=0∑∞1−j=0∑∞1=1+j=0∑∞(1−1)=1(1)(2)(3)(4)(5)(6)(7)
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