# How do you find 1/(1+2^-1)xx1/(1+3^-1)xx1/(1+4^-1)xx...xx1/(1+n^-1)?

Jan 20, 2016

${\prod}_{k = 2}^{n} \frac{1}{1 + {k}^{- 1}} = \frac{2}{n + 1}$

#### Explanation:

${\prod}_{k = 2}^{n} \frac{1}{1 + {k}^{- 1}} = {\prod}_{k = 2}^{n} \frac{k \cdot 1}{k \left(1 + {k}^{- 1}\right)}$

$= {\prod}_{k = 2}^{n} \frac{k}{k + 1}$

$= \frac{2 \cdot 3 \cdot 4 \cdot \ldots \cdot n}{3 \cdot 4 \cdot 5 \cdot \ldots \cdot \left(n + 1\right)}$

$= \frac{2}{n + 1}$

Jan 20, 2016

${\prod}_{k = 2}^{n} \left(\frac{1}{1 + {k}^{- 1}}\right) = \frac{2}{n + 1}$

#### Explanation:

$\frac{1}{1 + {k}^{- 1}} = \frac{k}{k + 1}$

So:

${\prod}_{k = 2}^{n} \left(\frac{1}{1 + {k}^{- 1}}\right) = \frac{2}{3} \times \frac{3}{4} \times \ldots \times \frac{n}{n + 1} = \frac{2}{n + 1}$

Jan 20, 2016

$s = \frac{2}{n + 1}$

Just a different style of writing the same thing as the others!

#### Explanation:

Write as:

Let the sum be $s$

$s = \frac{1}{1 + \frac{1}{2}} \times \frac{1}{1 + \frac{1}{3}} \times \frac{1}{1 + \frac{1}{4}} \times \ldots \times \frac{1}{1 + \frac{1}{n}}$

$s = \frac{1}{\frac{3}{2}} \times \frac{1}{\frac{4}{3}} \times \frac{1}{\frac{5}{4}} \times \ldots \times \frac{1}{\frac{n + 1}{n}}$

$s = \frac{2}{3} \times \frac{3}{4} \times \frac{4}{5} \times \ldots \times \frac{n}{n + 1}$

Numerator-> n!

Denominator->((n+1)!)/2

So we have:color(white)(..)s= (2n!)/((n+1)!)

s=(2cancel(n!))/(cancel(n!)(n+1))

$s = \frac{2}{n + 1}$