Here, we will look at numerical errors that occur due to the addition of small numbers and the subtraction

of similar numbers. Start with x $=1010$ and calculate R according to

R $=$

$$

$$x $+$

$1$

$$

j$=1$

$$

$$ x

for the values of $=0\mathrm{.}01,0\mathrm{.}001,...,1010.$ To compute the term in brackets, first compute x $+$ and then

add again to yield x $+2$ until you reach the final addition in the sum. Note: Do not attempt to store all

intermediate values!

$($a$)$ Analytically, what should the value of R be in each case?

$($b$)$ Implement this in Julia and plot R as a function of with the x$-$axis on a logarithmic scale.

$($c$)$ What happens as decreases? W