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1 MATH 348 - Homework II Hand in by Oct 9 Late fee = 10% per day. Some cause happiness wherever they go; others whenever they go. Oscar Wilde 1+x 1. (10 points) Find the series expansion for f (x) = ln , |x| < 1. Note: Finding the 1−x ∞ X f n (0) series using only the definition f (x) = xn is a hard road to follow. A given fact: n=0 n! using the formula, ln(2) = ∞ X ∞ X (−1)k+1 1 1 (−1)N +1 (−1)k+1 = 1 − + + ··· + · · · = SN + k 2 3 N k k=1 k=N +1 and the estimate | ln(2) − SN | < (1) 1 shows that ln 2 ≈ .69318 is accurate to five digits. N +1 (a) (3 points) How large is N in (1) to obtain the five digit estimate stated? (b) (4 points) Find the Maclaurin series for f (x). (c) (3 points) Pick an x in the series you found for f (x) in part (b) and evaluate SN until you have an approximation for ln(2) which is as accurate as was the approximation in part (a). How does this N compared to the N needed in part (a). n-factors z }| { 2. (8 points, 4 points each) The symbol (a)n = a(a + 1)(a + 2) · · · (a + n − 1) is called Pochhammer’s symbol. It is also called the ascending (or rising) factorial. For example, (2)2 = (2)(3) = 6 and (3)3 = (3)(4)(5) = 60. Show (i) (n)n = (2n − 1)! and (ii) (n)n = 2n−1 (2n − 1)!! (n − 1)! (2) where (2n−1)!! = (2n−1)(2n−3) · · · 1 could be called the odd factorial. Hint: For (i), use the definition of (a)n with a = n and multiply by an adroit form of the number one. For (ii) 1! use induction: n = 1 is clear (1)1 = 0! = 1 and use (ii) to show (n + 1)n+1 = 2n (2n + 1)!!. 3. (12 points) For the differential equation y 00 (x) + xy 0 (x) + y(x) = 0 (a) (4 points) Find the two (independent) series solutions y1 (x) and y2 (x). (b) (2 points) Show that one of these solutions, say y1 (x), is y1 (x) = e−x 2 /2 . (c) (2 points) Show that the other solution, call it y2 (x), can be written in ∞ X (−1)n 2n n! 2n+1 x . Hint: Use #2 above. the form y2 (x) = n=0 (2n + 1)! (d) (4 points) Now use the reduction of order formula (page 525) to find a second independent solution. Show that the first three terms of the series expansion of this solution agrees with the result in part (c).