Problem

$1)$

$$The Taylor series expansion of

ta

n

$-$

$1$

x

$$for

$|$x$|$

$<$

$1$

$$is:

ta

n

$-$

$1$

x

$=$

x

$-$

x

$3$

$3$

$+$

x

$5$

$5$

$-$

cdots

$=$

$$

n

$=$

$0$

$\backslash$infty

$($

$-$

$1$

$)$

n

x

$2$n

$+$

$1$

$2$n

$+$

$1$

Develop an algorithm that performs the following

$($

you can present it in the form of a pseudo

$-$

code or a list of steps

$)$

:

The algorithm determines

ta

n

$-$

$1$

x

$$using the above equation, where the argument

x

$$is the input, and the output argument

y

$$is the approximate value for

ta

n

$-$

$1$

x

$$obtained using Taylor's series expansion. If

a

n

$$is the

n

th term of the series, then sum

S

n

$$of the first

n

$$terms is

S

n

$=$

S

n

$-$

$1$

$+$

a

n

$.$

$$In each pass, calculate the estimated error

E

$$given by

E

$=$

$|$

a

n

$|$

$.$

$$Stop adding terms when

E

$<$

$10$

$-$

$9$

$,$

$$and report y