. If the nth term of the sequence < an > is given by"211 322 433· ·n + 1nn# 1nFind the limit of sequence

1. Break Down the Problem

We need to find the limit of the sequence defined by the nth term $a_n$, which is given as: $a_n = \frac{21}{1} \cdot \frac{32}{2} \cdot \frac{43}{3} \cdots \frac{(n+1)n}{n}$

2. Relevant Concepts

To analyze this sequence, we can express each term in a more simplified format. The general term appears to be: $\frac{(n+1)(n)(n-1)\ldots(2)}{n!}$ We need to simplify $a_n$ and investigate its behavior as $n$ approaches infinity.

3. Analysis and Detail

Let’s write down the nth term clearly: $a_n = \prod_{k=1}^{n} \frac{(k+1)k}{k} = \prod_{k=1}^{n} (k+1) = (n+1)!$ Thus, we have: $a_n = (n+1)!$ Next, we need to find the limit of $a_n$ as $n$ approaches infinity.

4. Verify and Summarize

We know that as $n$ approaches infinity, $(n+1)!$ grows without bound. Therefore, the limit can be mathematically stated as: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} (n+1)! = \infty$ Thus, the sequence diverges to infinity.

Final Answer

The limit of the sequence $a_n$ as $n$ approaches infinity is: $\infty$