Excercises for Series

  1. Remember (the statements for) the following: Recursive Sequences, $p$-Series, geometric series, integral test, telescoping sum, comparison test, limit comparison test, ratio test, divergence test, test for alternating series.

  2. MATH [MATH, integral test.]

  3. MATH[MATH integral test].

  4. MATH[MATHintegral test]

  5. MATH [Divergent, divergent test].

  6. MATH [$=\allowbreak 1,$ telescoping sum].

  7. MATH [MATHtest for alternating series; note that this series converges 'conditionally', not absolutely.]

  8. MATH [MATHtest for alternating series; you need to prove MATH is a decreasing function and its limit is $0.]$

  9. MATH [converges conditionally; prove that MATH is divergent by using limit comparison test with MATH

  10. Prove that MATHconverges conditionally.

  11. MATH[MATH use MATHand use the ration test on MATH

  12. MATH[convergent; use limit comparison test on MATH

  13. MATH

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