Homework $1$

Consider the following infinite series representations of $$ :

$\frac{}{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-$dots

$\frac{{}^2}{8}=1+\frac{1}{3^2}+\frac{1}{5^2}+\frac{1}{7^2}+\frac{1}{9^2}+$dots

In this HW$,$ you will write a PYTHON program to calculate $$ using the above expressions.

a$)$ For first evaluation retain $8$ significant digits in both cases, using the relative error percentage to assess

convergence. How many terms need to be retained in each case to obtain the desired result, i$.$e$.,$ which

expression converges faster?

b$)$ Repeat the calculations by calculating the true error instead of relative error, i$.$e$.,$ comparing your result at

each iteration with true value of $($which can be obtained using math.pi in PYTHON$).$ In this part, stop

the iterations when the absolute value of the true error becomes smaller than ${10}^{-7}.$ Comment on how the

results for this case compare with those in $($a$).$

c$)$ For $($b$),$ also plot the $($absolute$)$ true error as a function of iterations to compare the convergence of both

expressions. What do you observe in this plot?

What to submit:

Two separate PYTHON codes for part $($a$)$ and $($b$),$ which reproduce the results upon execution, i$.$e$.,$ print out

the iteration number, the partial sum and the error directly to the command line.

A very short writeup showing the plot in $($c$)$ and providing brief answers to the questions. The writeup should

absolutely not be more than one page.