For this exercise define
- Using the trig identity
twice, once with
, and then the second time with
, show that
Then use the same identity again with
and
to show
This establishes the identity
- Using the Taylor polynomial approximation
at
prove that
- Using the Taylor polynomial approximation
at
prove that
- Using the above parts show that the value of
to seven decimal places is 3.1415926.
- Proof. Letting
we have
Letting
we have
Letting
and
we have (recalling that
)
But then
- Proof. We know the Taylor polynomial approximation for
from this exercise (Section 7.8, Exercise #3):
Therefore, we can compute an approximation to
,
where
Therefore,
- Proof. Again using the Taylor polynomial approximation to
we have
- Finally,
should be, 16E_12(x) instead of E_10(x), check formula for T_11(atan(x)) on page 277, the sum ends with index n, which means in exercise 7.8.3 n is actually 6, so E_2n(x) is E_12(x).
Do we have to multiply errors by 16 and -4 here (since the values of arctan are being multiplied by 16 and -4)?
PS I meant in the question above (I think I replied to a comment when I didn’t mean to)