Discussion
For each positive integer \(n\), let \(x_n = 1 - (-1)^{n} + \frac{1}{n}\) and \(y_n = \left(1 + \frac{1}{2n}\right)^{3n}\).
Which ONE of the following statements about the sequences \((x_n)\) and \((y_n)\) is TRUE?
Answer: D
\((x_n)\) is not convergent and \((y_n)\) is convergent.
For even \(n\), \((-1)^{n} = 1\) and \(x_n = \frac{1}{n} \to 0\). For odd \(n\), \((-1)^{n} = -1\) and \(x_n = 2 + \frac{1}{n} \to 2\). Two subsequences with different limits mean \((x_n)\) does not converge.
For \((y_n)\), regroup the power so the standard limit \(\left(1 + \frac{1}{m}\right)^{m} \to e\) appears with \(m = 2n\):
\[ \begin{aligned} y_n &= \left[\left(1 + \frac{1}{2n}\right)^{2n}\right]^{3/2} \\ &\to e^{3/2} \end{aligned} \]
So \((y_n)\) converges. The options saying \((x_n)\) converges miss the alternating \((-1)^{n}\) term, which does not die out; the option saying \((y_n)\) diverges treats it like \((1 + \frac{1}{2n})^{n^{2}}\), where the exponent grows too fast.
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