MATH-0042 Study Guide - Midterm Guide: Partial Fraction Decomposition, Improper Integral, Indeterminate Form
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Solutions to math 42 second exam february 21, 2013: (10 points) determine, with justi cation, whether each series converges. Xn=2 (5 points) we see that n2 1 = (n + 1)(n 1), so we will use a partial fraction decomposition: We multiply through by (n + 1)(n 1) and group like terms: 1 = a(n 1) + b(n + 1) = an a + bn + b = (a + b)n + (b a)n. Thus, we have that a + b = 0 and b a = 1. Adding these two equations yields 2b = 1, so that b = 1. 2 and a = b = 1. Writing out the rst few partial sums, we notice telescoping: s2 = 1 s3 = 1 s4 = 1 s5 = 1 sn = 1. 2n(cid:19) (5 points) we will work with each part of the sum separately. This is a geometric series with r = 4.
