\[ 1 - \frac{1}{{\color{blue}2}} + \frac{1}{{\color{blue}2}} - \frac{1}{{\color{red}3}} +\frac{1}{{\color{red}3}} - \frac{1}{{\color{green}4}} + \frac{1}{{\color{green}4}} - \cdots = 1 \] This infinite telescoping sum converges to \(1,\) partly because all the terms except \(1\) get cancelled.

Can we say that the infinite telescoping product below **also** converges to \(1\) for a similar reason?

\[ \frac{1}{{\color{blue}2}} \times \frac{{\color{blue}2}}{{\color{red}3}} \times \frac{{\color{red}3}}{{\color{green}4}} \times \frac{{\color{green}4}}{{\color{brown}5}}\times \frac{{\color{brown}5}}{{\color{pink}6}} \times \frac{{\color{pink}6}}{7} \times {\cdots } = 1\]

×

Problem Loading...

Note Loading...

Set Loading...