1 Find the radius of convergence and interval of convergence of the series. ?_{n=2}^{?} (b)^n / ln(n) (x - a)^n, b > 0 2 Find the power series representation for the function and determine the radius of convergence. f(x) = (x / (2 - x))^3 3 Find the Maclaurin series for f(x) using the definition of a Maclaurin series. [Assume that f has a power series expansion. Do not show that Rn(x) ? 0.] Also find the associated radius of convergence. f(x) = cos(x) 4 Use Maclaurin series in Table 1 below (table is found in Chapter 11 Sec 10) to obtain the Maclaurin series for the given function. f(x) = sin(?x / 4) 5 Evaluate the indefinite integral as an infinite series. ? arctan(x^2) dx
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We have the series $\sum_{n=2}^{\infty} b^n (x-a)^n$. To find the radius of convergence, we can use the Ratio Test. Let's compute the limit: $$\lim_{n\to\infty} \frac{b^{n+1}(x-a)^{n+1}}{b^n(x-a)^n} = \lim_{n\to\infty} b(x-a) = b(x-a).$$ For the series to Show more…
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