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).