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# Compute π using the Taylor polynomial of arctan x

For this exercise define

1. 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

2. Using the Taylor polynomial approximation at prove that

3. Using the Taylor polynomial approximation at prove that

4. Using the above parts show that the value of to seven decimal places is 3.1415926.

1. Proof. Letting we have

Letting we have

Letting and we have (recalling that )

But then

2. 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,

3. Proof. Again using the Taylor polynomial approximation to we have

4. Finally,

### One comment

1. Anonymous says:

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).