4. (a) Find the MacLaurin Series for arctan(x) (up to the term in x^5) by explicitly calculating the first five derivatives of arctan. (b) Now use the geometric series approach to find a power series for arctan(x). [We did this in class in Ex 3.6.] Check that you get the same result as in (a). 5. By any method find an expression for the sum of the infinite series: f(x) = sum_{n=1}^{inf} x^{n+2} / (n(n+2)) = x^3 / (1 * 3) + x^4 / (2 * 4) + x^5 / (3 * 5) + x^6 / (4 * 6) + ... (-1 <= x <= 1) For any integrations you need to do you may use technology. 6. Use Taylor's inequality to decide how many terms of the series 1 + 1/(3 * 1!) + 1/(3^2 * 2!) + 1/(3^3 * 3!) + 1/(3^4 * 4!) + 1/(3^5 * 5!) + ... we need to take to approximate e^{1/3} to within 10^{-8}.
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The MacLaurin series is given by: $$f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \frac{f^{(4)}(0)}{4!}x^4 + \frac{f^{(5)}(0)}{5!}x^5 + \cdots$$ We need to find the first five derivatives of $arctan(x)$ and evaluate them at $x=0$. $$f(x) Show more…
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