Use the formula

Verify the Gregory–Leibniz formula in part (d) as follows.
(a) Set r = −x² in Eq. (6) and rearrange to show that

(c) Use the Comparison Theorem for integrals to prove that

Observe that the integrand is ≤ x^2N.

(d) Prove that

Use (b) and (c) to show that the partial sums S_N satisfy ΙS_N − π/4Ι ≤ 1/2N+1 , and thereby conclude that 

Transcribed Image Text:

1+r+²++N-1 1-N 1-r