Give an example of a bounded sequence that has a limit. Choose the correct answer below. A. \( a_{n}=(-1)^{(n+1)}, n \geq 1 \) B. \( a_{n}=\tan n, n \geq 1 \) C. \( a_{n}=\frac{e^{n}}{n}, n \geq 1 \) D. \( a_{n}=\frac{n}{n+1}, n \geq 1 \)
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The correct answer is D. \( a_{n}=\frac{n}{n+1}, n \geq 1 \). As \( n \) approaches infinity, this sequence approaches the limit of 1, making it bounded (by 0 and 1) and convergent. Now isn't that a neat way to see how sequences behave as they grow? One fun fact to remember about bounded sequences is that they don't just wander off into the infinity of numbers. They stay snug within certain limits, and some even have predictable patterns! So next time you're tackling sequences, think of them as little travelers on a defined path.