Telescoping : Product it or sum it

Algebra Level 1

112+1213+1314+14=1 1 - \frac{1}{{\color{#3D99F6}2}} + \frac{1}{{\color{#3D99F6}2}} - \frac{1}{{\color{#D61F06}3}} +\frac{1}{{\color{#D61F06}3}} - \frac{1}{{\color{#20A900}4}} + \frac{1}{{\color{#20A900}4}} - \cdots = 1 This infinite telescoping sum converges to 1,1, partly because all the terms except 11 get cancelled.

Can we say that the infinite telescoping product below also converges to 11 for a similar reason?

12×23×34×45×56×67×=1 \frac{1}{{\color{#3D99F6}2}} \times \frac{{\color{#3D99F6}2}}{{\color{#D61F06}3}} \times \frac{{\color{#D61F06}3}}{{\color{#20A900}4}} \times \frac{{\color{#20A900}4}}{{\color{#624F41}5}}\times \frac{{\color{#624F41}5}}{{\color{#E81990}6}} \times \frac{{\color{#E81990}6}}{7} \times {\cdots } = 1

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