Calc BC Mid-Unit 10 CA

Infinite Sequences and Series Mid-Unit 10 Corrections

1. Infinite Series and Partial Sums

  • The given infinite series has its $n$th partial sum denoted as $S_n = …$.
  • Inquiry on the full sum of the series:
    • The method to compute the sum was not specified in the transcript; generally requires evaluation techniques or limits.

2. Diverging Series

  • Identification of which series diverge:
    • Options provided: I, II, III
    • Correct answer: (C) III only

3. n-th Term Test for Divergence

  • The $n$th-Term Test is applicable for considering divergence of:
    • Options: I, II, III
    • Correct answer: (C) I and III only.

4. Sum Calculation with Real Numbers

  • If $b$ and $t$ are real numbers where $0 o |t| o |b|$, the sum is to be assessed:
    • Specific details on $b$ and $t$ values and their relationship were not included.

5. Integral Test Applicability

  • Explanation required for why the Integral Test does not apply to a specific series.
  • Detailed reasoning needed:
    • Typically, the conditions include whether the function linked to the series is monotonic or continuous over the interval in question.

6. Convergence Values for Infinite Series

  • Investigation into which values of $p$ will ensure the convergence of the infinite series:
    • Details were not elaborated in the transcript; refer to the $p$-series test where convergence occurs if $p > 1$.

7. Matching Series from Partial Sums Sequence

  • The partial sums sequence is:
    • $0.1667, 0.3333, 0.4833, 0.6167, 0.7357, …$
    • Given options for a matching series are: (A), (B), (C), (D).
    • Correct option determined: (B).

8. Conditional Convergence Values for Series

  • Inquiry regarding at what values of $x$ does the series remain conditionally convergent:
    • Provided choices for $x$:
    • (A) $x = …$
    • (B) $x = …$
    • (C) $x = …$
    • (D) $x = …$
    • Correct selection: (A).

9. Limit Comparison Test Series

  • Evaluation of which series is suitable for the Limit Comparison Test regarding convergence/divergence:
    • Offered choices: (A), (B), (C), (D).

10. Divergence Verification Using n-th Term Test

  • Task to verify divergence using the $n$th-Term Test:
    • Show the value of the limit.
    • Note: The series diverges, limit computation details required also.

11. Statements About a Given Series

  • A series defined as ∑ $a_n$. Investigation points:
    • (A) The series diverges by the $n$th Term Test.
    • (B) The series diverges by comparison with …
    • (C) The series converges by comparison with …
    • (D) The series converges by comparison with …
    • Truthful statement determined: (C).

12. Convergence via Alternating Series Test

  • Series identification that converges via the Alternating Series Test:
    • Given series I, II, III
    • Correct response:
    • A. I only
    • B. I and II only
    • C. I and III only
    • D. I, II, and III
    • Chosen answer: (B).

13. Absolute Convergence of Series

  • Series assessed for absolute convergence: I, II, III
    • Correct options identified:
    • (A) I only
    • (B) I and II only
    • (C) I and III only
    • (D) I, II, and III
    • Final answer selected: (B).

14. Convergence/Divergence via Integral Test

  • Use of the Integral Test to assert convergence or divergence of the series:
    • Computation of $ ext{∫} f(x) ext{dx} o ext{∞}$ led to conclusion that the series diverges.

15. Statements Validity Regarding a Series

  • Series investigation concerning:
    • I. $a_n = a_n$ for all $n o 1$.
    • II. $ ext{lim}_{n o ext{∞}} a_n = 0.$
    • III. The series converges by the Alternating Series Test.
    • Correct selections:
    • A. I only
    • B. I and II only
    • C. II and III only
    • D. I, II, and III
    • Answer selected: (B).

16. Value Determination for Convergence of Series

  • Elucidate on all values of $x = 0$ for which the series converges.

17. Identification of Convergent p-Series

  • Selection of the convergent $p$-series:
    • Given options (A), (B), (C), (D).

18. Analysis of a Series

  • Consider the series $a_n$:
  • Condition given: $a_n$ is even for all integers $n = 1$.
  • Required value: What is $a_n$?

Answers to Mid-Unit 10 Corrective Assignment

  1. ...
  2. C
  3. C
  4. ...
  5. f_n is not a decreasing function for $n = 1$.
  6. $p = 0$
  7. B
  8. A
  9. B
  10. Diverges by $n$th-Term Test, $ ext{lim}_{n o ext{∞}} a_n = … $
  11. C
  12. B
  13. B
  14. $ ext{∫} f(x) o ext{∞}$, Series Diverges
  15. B
  16. $x = 7$
  17. D
  18. 32$…$