Inspired by Aditya Sharma

Calculus Level 4

In which step did we first make a mistake?

0=i=00(1)=i=0(11)(2)=i=01i=01(3)=(1+i=11)j=01(4)=1+j=01j=01(5)=1+j=0(11)(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}


Inspiration.

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