(a) Find the Maclaurin series for the function f(x)= 1 f(x) = - f(x)= 1+x. 1...

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(a) Find the Maclaurin series for the function f(x)= 1 f(x) = - f(x)= 1+x². 1 X - 1 1 + x (b) By integrating both sides of the Maclaurin series for 1+x²³ show that the Maclaurin series for the function f(x)= arctanx is x² x²n+1 - + ... = [(-1)"+¹ (c) Using the Maclaurin series for 7 2n+1 up to and including the term with x¹³, show that (c) Show that the exact value of the integral is x³ x³ + 3 5 f(x) = arctan x 1/√3 | arctan.x dx = 0.158459. 1/√√3 farctan x dx = 1 π Do 1 - and hence that for In 2 4 . (d) Hence deduce that an approximate value of 3 √36 = is π 3.14159. (e) To how many decimal places is this approximation expected to be accurate? (a) Find the Maclaurin series for the function f(x)= 1 f(x) = - f(x)= 1+x². 1 X - 1 1 + x (b) By integrating both sides of the Maclaurin series for 1+x²³ show that the Maclaurin series for the function f(x)= arctanx is x² x²n+1 - + ... = [(-1)"+¹ (c) Using the Maclaurin series for 7 2n+1 up to and including the term with x¹³, show that (c) Show that the exact value of the integral is x³ x³ + 3 5 f(x) = arctan x 1/√3 | arctan.x dx = 0.158459. 1/√√3 farctan x dx = 1 π Do 1 - and hence that for In 2 4 . (d) Hence deduce that an approximate value of 3 √36 = is π 3.14159. (e) To how many decimal places is this approximation expected to be accurate?

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