Examination Instructions

The examination document is structured with specific instructions and details regarding the examination format, subjects, and regulations. The document contains the following key components:

Basic Guidelines

• This paper is designated for use during the examination; candidates must not remove it from the examination hall.
• All papers and answer booklets should be placed in the exam envelope provided.
• Faculty: Natural, Mathematical & Engineering Sciences
• Department: Mathematics
• Mock Exam Year: 2026
• Student Candidate Number: [To be filled by the student]
• Seat Number: [To be filled by the student, including color zone if applicable]
• Module Code: 4CCM115A
• Module Title: Sequences and Series
• Time Allowed: Two hours
• Total Marks: 100
• Total Pages: 8
• Total Questions: 12
• Permitted Materials: Candidates may not bring books, notes, or other written materials; calculators are also prohibited.

Examination Structure

• The paper is divided into Section A and Section B.
• Both sections contribute equally to the total marks, with Section A and Section B each accounting for 50%.
• Instructions for candidates: Answer ALL questions.
• The examination follows the regulations set by the Academic Board of the institution, and copying or sharing the exam paper is strictly prohibited.

Section A

Section A contains ten questions (A1 to A10). Each question is worth 5 marks if answered correctly; incorrect answers will incur a penalty of -1 mark, and blank or invalid answers receive 0 marks.

Question Breakdown

A1

• Statement Analysis:
    - Evaluate which statement is NOT true:
      - A) $\bigcup_{n=1}^{ ext{∞}} [-n, n) = ext{R}$
      - B) $\bigcup_{n=1}^{ ext{∞}} (1, 2 - \frac{1}{n}) = ext{∅}$
      - C) $[-2, 2] \backslash (-3, 1) = [1, 2]$
      - D) $\bigcup_{n=1}^{ ext{∞}} [ ext{√}n, n] = [1, ext{∞})$
      - E) $\bigcup_{n=1}^{ ext{∞}} (0, 1 + \frac{1}{n}) = [0, 1]$

A2

• Proposition Negation: Given the proposition about the subsets $X$ and $Y$, ascertain the negation:
    - Proposition: orall x ext{ in } X, ext{ there exists } y ext{ in } Y ext{ such that } x^2 ext{ is less than or equal to } y < 2x
    - Options for negation:
      - A) ext{∃ } x ext{ in } X, ext{ }∀ y ext{ in } Y, ext{ } y < x^2 ext{ or } y ext{ }
  ot ext{< } 2x
      - B) $ext{∃ } x <br />\not ext{in } X, ext{ }∀ y ext{ in } Y…$
      - C) $ext{∃ } x ext{ in } X, ext{ }∀ y ext{ in } Y, …$ (further details may continue)

A3

• Set Properties: Analyze the set X = iggrac{n + (-1)^n}{n^2 + 1} : n ext{ in } ext{N} ackslash ext{1} for truth statements:
      - A) If $x ext{ in } X$, then x < 1
      - B) If $x ext{ in } X$, then $x ext{ } ≥ 0$
      - C) $X$ has a minimum element.
      - D) $ext{sup}(X) = \frac{3}{5}$.
      - E) $X$ has a maximum element.

A4

• Convergence Conditions: Determine which condition does NOT imply that a_n
  ightarrow 0 ext{ as } n
  ightarrow ext{∞}:
      - A) If a_n^3
  ightarrow 0
      - B) If a_n ext{ is such that } a_n ext{ ≤ } 0 ext{ and } a_{n+1} > a_n
      - C) If the sequence is bounded.
      - D) If orall ext{ε} > 0, ext{ } ext{∃ } n_0 ext{ such that } orall n ≥ n_0, |a_n| < ext{√ε}
      - E) If ext{∃ } n_0 ext{ such that } orall n ≥ n_0, |a_n| < rac{1}{ ext{√}n}

A5

• Limit Evaluation: Analyze the limit of the sequence defined as:
  $a_n = \frac{(-3)^{n+2} - 3n n^3}{3n + 1 n^3 + 3 }$ to determine the correct limit:
      - A) $ext{lim}_{n o ext{∞}} a_n = 3$
      - B) Other values until E)

A6

• Determine the truth of the following sequences:
      - A) The sequence $a_n = \frac{3n}{n!}$ is increasing.
      - B) The sequence $a_n = \frac{n-1}{n+1}$ is not monotonic.
      - C) The sequence $a_n = \frac{n^2}{2n}$ is increasing.
      - D) The sequence $a_n = n^2 - 10n$ is not monotonic.
      - E) None of the other options is correct.

A7

• For a sequence with two limit points, assess various statements for their truth:
     - A) If the limit points are $l_1$ and $l_2$ (where $l_1 = -l_2$), then the sequence of absolute values converges.
     - B) The sequence does not converge.
     - C) The sequence is not monotonically increasing.
     - D) If the absolute value sequence has unique limit point, then $l_1 = -l_2$.
     - E) The sequence $ext{-}a_n$ has precisely two distinct limit points.

A8

• Evaluate properties of bounded sequences:
     - A) Every monotonic subsequence converges.
     - B) For every ext{ε} > 0, there exists $n_0$ such that for all m,n > n_0, $|a_m - a_n| < \frac{210 ext{ε}}{1}$.    - C) Existence of a subsequence that is a Cauchy sequence.    - D) There exists some real number $l$, where for all ε>0 and n in N, ext{∃} m > n, |a_m - l| < ext{ε}.
     - E) The sequence $a_n^2 - 2a_n$ is bounded.

A9

• Identify convergence for the series igg ext{∞ }_{k=1} a_k:
     - A) $a_k = \frac{1}{ ext{√}k}$
     - B) $a_k = \frac{k}{k + ext{√}k}$
     - C) $a_k = \frac{(-1)^k}{ ext{√}k}$
     - D) $a_k = \frac{ ext{√}k}{1 + ( ext{log} k)^2}$
     - E) $a_k = \frac{1}{k} - \frac{1}{k^2}$

A10

• Evaluate the truth in the behavior of the series igg ext{∞ }{k=1} a_k based on different conditions:    - A) If the series of squares converges, the series igg ext{∞ }{k=1} a_k must also converge.
     - B) If igg ext{∞ }{k=1} a_k converges, then igg ext{∞ }{k=1} (-1)^{k+1} a_k must also converge.
     - C) If the series of absolute values converges, then the original series must converge.
     - D) If a_k
  ightarrow 0 ext{ as } k
  ightarrow ext{∞}, this does not guarantee convergence.
     - E) If the series converges, absolute convergence must follow.

Section B

In Section B, students are not permitted to use theorems from the course and must address questions directly from definitions and proofs.

Question Breakdown for Section B

B11

• Convergence of Sequences: State the definitions for:
    - Convergence of a sequence of real numbers $ext{{a_n}}$.
    - Bounded below and bounded above.
    - Limit Point: Define a limit point and provide an example of an unbounded sequence with three limit points.

B12

• Supremum and Infimum: Define the supremum (sup) and infimum (inf) of a bounded set of real numbers.
    - For set X := iggrac{(-1)^n + 1}{2n}: n ext{ in } N, state the inf and sup without proof, then provide proof for the stated infimum.

• Bounded Set Transformation: Prove that if $X$ is bounded and $ext{λ}, ext{µ}$ are real numbers (with ext{λ} > 0), then set Y := igg ext{λ} x + ext{µ}: x ext{ in } X is bounded and $ext{sup}(Y) = ext{sup}(X) λ + μ$.

• Series Convergence: Define convergence for series $ext{∑}<em>{n=1}^{∞} a_n$ and absolute convergence. State the alternating series test and use it to verify the convergence of the series $ext{∑}</em>{n=1}^{∞} (-1)^{n+1} \frac{1}{n}$, showing that it does not converge absolutely.