गणित एवं संख्यात्मक विश्लेषण

Evaluation of sequence limit:

• Incorrect: The limit is 0, not 1.
• Correct: Taking the limit as $n \to \infty$:
  $$\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{n^2} = 0$$
  Because a finite unique limit exists ($L = 0$), the sequence $(a_n)$ is convergent and its limit is 0.
• Incorrect: The sequence is bounded and monotonically decreasing to 0, hence convergent.
• Incorrect: A convergent sequence in $\mathbb{R}$ possesses a unique limit, never multiple limits.

Step-by-step analysis of each option for $a_n = \frac{1}{\sqrt{n}} = n^{-1/2}$:

• Incorrect: By the Alternating Series (Leibniz) Test, since $a_n \gt 0$, $a_n$ is monotonically decreasing to $0$, $\sum (-1)^n a_n$ converges (conditionally).
• Incorrect: $\sum (a_n)^2 = \sum \frac{1}{n}$, which is the divergent harmonic series ($p = 1$).
• Correct: $\sum_{n=1}^\infty (a_n)^3 = \sum_{n=1}^\infty \left(\frac{1}{\sqrt{n}}\right)^3 = \sum_{n=1}^\infty \frac{1}{n^{3/2}}$. This is a standard $p$-series with $p = 3/2 \gt 1$, which is convergent.
• Incorrect: $\sum a_n = \sum \frac{1}{n^{1/2}}$ is a $p$-series with $p = 1/2 \le 1$, which is divergent.

Analysis of convergence theorems in real analysis:

• Correct: A fundamental theorem states that if a series is absolutely convergent (i.e., $\sum |a_n|$ converges), then the series $\sum a_n$ itself is necessarily convergent.
• Incorrect: The converse is false; for example, $\sum \frac{(-1)^n}{n}$ converges but is not absolutely convergent.
• Incorrect: Conditional convergence means the series converges but does NOT converge absolutely, which is the exact opposite.
• Incorrect: Conditionally convergent series can never be absolutely convergent by definition.