Vollständige Induktion (1-1/2)*(1-1/3)*...*(1-1/n) = 1/n

Question

Kann mir jemand hierbei helfen:

Answer 1

$$ \prod_{i=2}^n \left( 1 - \frac{1}{i} \right) $$

$$ \rightarrow \quad \prod_{i=2}^{n+1} \left( 1 - \frac{1}{i} \right) = \prod_{i=2}^n \left( 1 - \frac{1}{i} \right) \color{red}{ \cdot \left( 1 - \frac{1}{n+1} \right) } $$

$$  = \frac{1}{n} \color{red}{ \cdot \left( 1 - \frac{1}{n+1} \right) } = \frac{1}{n} - \frac{1}{n^2+n} = \frac{n^2+n}{n(n^2+n)} - \frac{n}{n(n^2+n)}$$

$$ \frac{n^2}{n(n^2+n)} = \frac{n^2}{n^2(n+1)} = \frac{1}{n+1}$$