Practice Test 3 — Detailed Solutions

GRE Math Subject · Axiom Academy

LESSON Practice Test 3 — Detailed Solutions Step-by-step solutions for 16 problems: series, ring theory, metric spaces, probability, ODEs, and complex analysis Determine whether ... converges or diverges. Use the identity: ... for |x| . Apply the integral test. Let u = ln x , du = dx/x : The integral converges, so the series converges. For which values of x does ... converge? Ratio test gives radius R = 4, center c = 2. So the open interval of convergence is (2-4, 2+4) = (-2, 6). Now check each endpoint. At x = 6: sum 1/n , the harmonic series, which diverges. At x = -2: sum (-1)^n/n converges by the alternating series test. So the interval of convergence is -2 , which is choice (C). If sum a_n converges absolutely and |b_n| for all n, does sum a_n b_n converge absolutely? Since |a_n b_n| and M*sum|a_n| , the comparison test gives absolute convergence of sum a_n b_n . In the ring Z[i] (Gaussian integers), which of the following is a unit? A unit in Z[i] has norm 1. The norm is N(a+bi) = a^2+b^2 . N(-i) = 1 -- unit! (inverse is i, since (-i)(i) = 1 ) The units of Z[i] are exactly 1, -1, i, -i . R[x]/(x^2+1) = C , which is a field. An ideal I is maximal if and only if R/I is a field. Since C is a field, (x^2+1) is maximal. Equivalently, x^2+1 is irreducible over R (no real roots), and R[x] is a PID, so irreducible elements generate maximal ideals. How many elements does the ring Z_2[x]/(x^3+x+1) have?

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