Course Landing Page

Series practice list handout - solutions

7 min read

List 1

Problem Determine whether each of the series is absolutely convergent, conditionally convergent, or divergent, using any convergence test.

  • (1)
  • (2)
  • (3)
  • (4)
  • (5)
  • (6)
  • (7)
  • (8)
  • (9)
  • (10)
  • (11)
  • (12)
  • (13)

Solution (1)

  1. && Use ratio test to determine convergence.
  2. & Derive conclusion.
    • Since , the series absolutely converges.

(2)

  1. && Simplify series.
  1. & Use alternating series test to determine convergence.
  1. & Derive conclusion.
    • Since and is a decreasing sequence, converges.
  2. & Determine absolute convergence.
    • Since , is a divergent -series, is conditionally convergent.

(3)

  1. & Set up limit comparison test to determine convergence. Compare series with
  2. & Determine comparison of using comparison test with .
    • Note that .
    • Since is a divergent -series, we conclude also diverges.
  3. && Perform limit comparison test.
  1. & Derive conclusion.
    • Since the above limit does not evaluate to zero, and our comparison series diverges, our given series diverges as well.

(4)

  1. & Use alternating-series test to determine convergence.
  1. & Derive conclusion.
    • Since the above limit does not evaluate to zero, we conclude our series diverges.

(5)

  1. && Use the ratio test to determine convergence.
  2. & Derive conclusion.
    • Since the above limit is greater than 1, the series diverges. (6)
  3. && Use ratio test to determine convergence.
  4. & Derive conclusion.
    • Since the above limit is greater than 1, the series diverges.

(7)

  1. && Use ratio test to determine convergence.
  2. & Derive conclusion.
    • Since the above limit is less than 1, the series absolutely converges.

(8)

  1. & Use alternating series test to determine convergence.
  1. & Derive conclusion.
    • Since the above limit evaluates to 0 and is a decreasing sequence, the given series converges.
  2. & Determine absolute convergence.
    • Since is a divergent -series, we concluded our given series is conditionally convergent.

(9)

  1. & Use root test to determine convergence.
  1. & Derive conclusion.
    • Since the above limit is less than 1, the given series converges absolutely.

(10)

  1. & Use root test to determine convergence.
  1. & Derive conclusion.
    • Since the above limit is less than 1, the given series converges absolutely.

(11)

  1. & Set up alternating-series test to determine convergence.
  1. & Derive conclusion.
    • Since the above limit evaluates to 0 and is a decreasing sequence, the series converges.
  2. && Determine absolute convergence using limit comparison test.
    • Compare with .
    • Since is a converging -series, also converges.
  3. & Derive conclusion.
    • Since converges, our given series is absolutely convergent.

(12)

  1. & Set up alternating series test to determine convergence.
  1. & Derive conclusion.
    • Since the above limit evaluates to 0 and is a decreasing sequence, the series converges.
  2. && Determine absolute convergence using comparison test.
    • Use for comparison, and not that .
    • Since is a divergent -series, also diverges.
  3. & Derive conclusion.
    • Since diverges, conditionally converges.

(13)

  1. && Use the integral test to determine convergence.
  1. & Derive conclusion.
    • Since the above integral converges, our given series is convergent.

List 2

Problem For each series, state a convergence test that will show whether is converges or diverges.

  • (43)
  • (44)
  • (45)
  • (46)
  • (47)
  • (48)
  • (49)
  • (50)
  • (51)
  • (52)
  • (53)
  • (54)
  • (55)
  • (56)

Solution (43)

  1. && Note that the series is a sum of two geometric series. Therefore, you can use the geometric series test.

(44)

  1. & Since the series involves a factorial, you should use the ratio test.

(45)

  1. & Since the series involves a polynomial over an power, the ratio test would be the best option.

(46)

  1. & Since the function is easily integrated, positive, and decreasing, the integral test is a good option.

(47)

  1. & The limit comparison test with is a good option for determining convergence. Note that the direct comparison test would not work since our comparison series converges and the given series is greater.
  2. & If you are comfortable with integrals involving trigonometric substitutions, the integral test can be used as well.

(48)

  1. & The direct comparison test with is an efficient way for determining convergence since the terms of the given series are less than those of the comparison series.

(49)

  1. & The -series test is the easiest way to determine convergence.

(50)

  1. & The divergence test is the easiest way to tell the series diverges.

(51)

  1. & This series can be rewritten as a geometric series, so the geometric series test can be applied.

(52)

  1. & The presence of a term implies using the alternating-series test along with the -series test to analyze the absolute value of the series.

(53)

  1. & The limit comparison test with is the best option.

(54)

  1. & The alternating-series test used with a limit comparison test with is the best option.

(55)

  1. & The ratio test is the best option since we have an power over a polynomial.

(56)

  1. & The root test is the best option here since all terms are raised to the power of .