a$)$ Create an array of integers running from $1$ through to n using a numpy function, and print the result. You choose the value of n$.$

b$)$ Create two n $\backslash$times n matrices: one in which each row is your array of integers created in part a$)$; the other where each column is your array. Name one $2$D array numerator and the other denominator. Note, the numpy function tile may be helpful here. Consider also the transpose operation.

c$)$ Set up an error matrix to compare fractions $\frac{i}{j}($where i and $j$ independently run from $1$ to $n)$:

err$=|\frac{i}{j}-|$

and return the smallest error when $n=100.$

d$)$ Use argmin and un ravel$\_$index to find the location of the minimum in your $2$D array, and with this find the values of the numerator and the denominator. Example usage from numpy manual:

ind $=$ np$.$unravel$\_$index$($np$.$argmin$(a,$ axis$=$None$),$ a$.$ shape$)$

e$)$ Write down the best approximations to z when n $=100,$ n $=1000$ and n $=10000.$

f$)$ Using the series formula provided in lectures:

$=(8(1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+...+\frac{1}{n^2}+...){)}^{\frac{1}{2}}$

to obtain a value for $$ with $n$ ranging from $1$ to $101$ and plot these values versus the number of terms in the series. Include on your plot a horizontal

dashed line showing the value of $$ given by your best fractional approximation in part e$).$