Subject GRE-Style: Sequence Convergence Proof

GRE Math Subject · Axiom Academy

EXAMPLE Subject GRE-Style: Sequence Convergence 3 worked problems on sequences, series convergence, and finding limits Key Tests: Ratio test, root test, comparison test, alternating series test. A sequence converges if it is bounded and monotone (Monotone Convergence Theorem). Question: Let and . If the sequence converges, what is ? If the sequence converges to , then taking limits on both sides of : So or . Since every 0"> (square roots are non-negative), we must have . Why it converges: The sequence is increasing (check 1 = a_1"> and induct) and bounded above by 2 (if then ). By the Monotone Convergence Theorem, it converges. Question: For which values of does converge? (A) 0"> (B) 1"> (C) (D) 2"> (E) All Apply the integral test. Let , so : This is a -integral in . It converges if and only if 1"> . GRE Tip: The "Cauchy condensation" hierarchy to memorize: diverges, diverges, converges. The boundary is always at at each "level." Question: Determine whether converges or diverges. (A) Converges absolutely (B) Converges conditionally (C) Diverges (D) Ratio test inconclusive (E) Depends on starting index Ratio Test with Stirling's Flavor Apply the ratio test. Compute: Since , the ratio test gives convergence. Key Pattern: Whenever you see vs. or similar, the ratio test almost always works cleanly, and is the identity that makes it resolve. This limit appears constantly on the GRE.

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