(1) Find the radius of convergence for the following series. Show your work.
and interval P∞ 2n 1 n (a) (x − 3) n=1 4 n−1 P∞ (−1) n √ (b) n=1 3n n (x + 2) P∞ n n! (c) n=1 10n n3 (x + 4) (2) Find the function P∞ f (x) whose power series is given in each problem below: (a) f (x) =P n=1 nxn ∞ 1 (b) f(x) = n=1 n+1 xn P∞ (c) f (x) = n=0 an xn where ( 3 if n is divisible by 5 an = 1 otherwise (3) Suppose f (x) = ∞ X kxk k=0 and g (x) = ∞ X k (−1) xk k=0 Find the first first three nonzero terms of the series for the product of the two functions, f (x) g (x). (4) Find the MacLaurin series for f (x) = x3 − 3x2 + 2x + 5 centered at x = 2. (5) Find the power series centered at x = 0 for the following. I would recommend not using the formula for Taylor/MacLaurin series, but manipulating series you already know. 1 (a) f (x) = 1+x P∞3 1 (b) f (x) = k=0 8+x 3 5 (c) f (x) = x2 +x−6 (hint: use the method of partial fractions) x (d) f (x) = 1−cos x2 (6) Suppose that ∞ X k3 k f (x) = (x − 5) k! k=0 Find f (5) (the 100th derivative of f at 5) (7) Use series and/or asymptotic equivalence to evaluate the limit √ 8 1 + x − 8 − 4x + x2 lim x→0 x − sin x Do not use L’Hospital’s rule. (8) (a) Use the binomial theorem to write the first four nonzero terms of the Maclaurin series for Z xp 3 8 + x3 dx (100) 0 (you do not have to simplify numerical expressions) (b) Suppose we want to estimate Z 0.1 p 3 8 + x3 dx 0 How many terms of the series from (a) should we add together to ensure that our sump is within 10−10 = 0.0000000001 of the correct value? 1 
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