# How do you prove that the 8-sd approximation to the value of the infinite continued fraction 0.0123456789/(1+0.0123456789/(1+0.0123456789/(1+...))) =0.012196914 ?

##### 2 Answers
Jul 8, 2016

Express as a solution of a quadratic, applying the quadratic formula to find value:

$\frac{- 1 + \sqrt{1.0493827156}}{2} \approx 0.012196914$

#### Explanation:

Let $k = 0.0123456789$.

We want to find the value of $x$ given by:

$x = \frac{k}{1 + \frac{k}{1 + \frac{k}{1 + \ldots}}}$

Then:

$\frac{k}{x} = k \div \left(\frac{k}{1 + \frac{k}{1 + \frac{k}{1 + \ldots}}}\right) = 1 + \frac{k}{1 + \frac{k}{1 + \frac{k}{1 + \ldots}}} = 1 + x$

Multiplying both ends by $x$ and transposing, we get:

$x + {x}^{2} = k$

Subtract $k$ from both sides and rearrange slightly to get:

${x}^{2} + x - k = 0$

Using the quadratic formula:

$x = \frac{- 1 \pm \sqrt{1 + 4 k}}{2}$

Since $k > 0$, we require $x > 0$ too. So the only suitable root of this quadratic is:

$x = \frac{- 1 + \sqrt{1 + 4 k}}{2}$

$= \frac{- 1 + \sqrt{1 + \left(4 \cdot 0.0123456789\right)}}{2}$

$= \frac{- 1 + \sqrt{1.0493827156}}{2}$

$\approx 0.01219691418437889686$

$\approx 0.012196914$ to $8$ s.d.

Jul 8, 2016

See below

#### Explanation:

It is an infinite expansion
$y = \frac{x}{1 + \frac{x}{1 + \frac{x}{1 + \frac{x}{\cdots}}}}$ so

$y = \frac{x}{1 + y} \to x = y \left(1 + y\right)$

but $y = 0.012196914$

then

$x = 0.012345678711123395$

but

${x}_{0} = 0.0123456789$

then $\left\mid {x}_{0} - x \right\mid = 1.88877 \cdot {10}^{-} 10$ within the required precision