Question

Consider the statements:
I. If a series of positive terms is not convergent, then it is either divergent or oscillatory.
II. In an alternating series, if $lim⁡n→∞∣un∣≠0\lim_{n \to \infty} |u_n| \neq 0$, then it is convergent.
III. In a series of positive terms, if $lim⁡n→∞un=0\lim_{n \to \infty} u_n = 0$, then it may be convergent or divergent.
IV. If $∑∣un∣\sum |u_n|$ is divergent and $∑un\sum u_n$ is convergent, then $∑un\sum u_n$ is conditionally convergent.

Which of the above statement(s) is(are) correct?

• A. All the statements are correct
• B. Statements I and III are correct
• C. Statements III and IV are correct
• D. Statements II, III and IV are correct

Hint:

• Series of positive terms If not convergent, it diverges to $+\infty$. It cannot oscillate finitely.
• $n$-th term test for divergence For any series $\sum a_n$, if $\lim_{n\to\infty} a_n \neq 0$ or the limit does not exist, the series diverges.
• Necessary condition for convergence If $\sum a_n$ converges, then $\lim_{n\to\infty} a_n = 0$. The converse is not always true (e.g., harmonic series).
• Conditional convergence A series $\sum u_n$ converges conditionally if $\sum u_n$ converges AND $\sum |u_n|$ diverges.
• Absolute convergence A series $\sum u_n$ converges absolutely if $\sum |u_n|$ converges. Absolute convergence implies convergence.

Answer 1

The Correct Option is C

Understanding the Problem:

• We are given four statements (I-IV) about the convergence of series.
• We need to determine which of these statements are correct.
• The options provided are combinations of these statements.

Analyzing Each Statement:

Statement I:

• "If a series of positive terms is not convergent, then it is either divergent or oscillatory."
• For a series of positive terms, if it is not convergent, it must diverge to infinity (since the partial sums are increasing and unbounded). Oscillatory behavior is not possible for positive terms.
• Conclusion: This statement is incorrect because a series of positive terms cannot be oscillatory.

Statement II:

• "In an alternating series, if \(\lim_{n \to \infty} |u_n| \neq 0\), then it is convergent."
• For an alternating series to converge, the limit of the absolute value of its terms must be zero (\(\lim_{n \to \infty} |u_n| = 0\)). If this condition is not met, the series cannot converge.
• Conclusion: This statement is incorrect because the condition \(\lim_{n \to \infty} |u_n| = 0\) is necessary for convergence.

Statement III:

• "In a series of positive terms, if \(\lim_{n \to \infty} u_n = 0\), then it may be convergent or divergent."
• For a series of positive terms, \(\lim_{n \to \infty} u_n = 0\) is necessary but not sufficient for convergence. For example, the harmonic series \(\sum \frac{1}{n}\) diverges, while \(\sum \frac{1}{n^2}\) converges.
• Conclusion: This statement is correct.

Statement IV:

• "If \(\sum_{n} |u_n|\) is divergent and \(\sum_{n} u_n\) is convergent, then \(\sum_{n} u_n\) is conditionally convergent."
• Conditional convergence is defined as a series converging while its absolute series diverges.
• Conclusion: This statement is correct.

Evaluating the Options:

• Option 1: All statements are correct → False (Statements I and II are incorrect).
• Option 2: Statements I and III are correct → False (Statement I is incorrect).
• Option 3: Statements III and IV are correct → True.
• Option 4: Statements II, III, and IV are correct → False (Statement II is incorrect).

Final Answer:

Option 3: Statements III and IV are correct.