The problem asks to determine whether the sequence $a_n = \left(\frac{n+1}{2n}\right)\left(1 - \frac{1}{n}\right)$ converges or diverges. If it converges, we need to find the limit.

Step 1: Simplify the expression

We begin by simplifying $a_n$: $a_n = \left(\frac{n+1}{2n}\right)\left(1 - \frac{1}{n}\right)$

Let's simplify each part:

• $\frac{n+1}{2n} = \frac{n}{2n} + \frac{1}{2n} = \frac{1}{2} + \frac{1}{2n}$
• $1 - \frac{1}{n}$ is already simplified.

Now substitute these into the expression for $a_n$: $a_n = \left(\frac{1}{2} + \frac{1}{2n}\right)\left(1 - \frac{1}{n}\right)$ Expanding this product: $a_n = \left(\frac{1}{2} + \frac{1}{2n}\right)\left(1 - \frac{1}{n}\right) = \frac{1}{2}\left(1 - \frac{1}{n}\right) + \frac{1}{2n}\left(1 - \frac{1}{n}\right)$

Now, expand each term: $a_n = \frac{1}{2} - \frac{1}{2n} + \frac{1}{2n} - \frac{1}{2n^2}$

Simplifying the expression: $a_n = \frac{1}{2} - \frac{1}{2n^2}$

Step 2: Find the limit

Now, we evaluate the limit of $a_n$ as $n \to \infty$: $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \left(\frac{1}{2} - \frac{1}{2n^2}\right) = \frac{1}{2} - 0 = \frac{1}{2}$

Conclusion

The sequence $a_n$ converges, and its limit is: $\lim_{n \to \infty} a_n = \frac{1}{2}$

Would you like further details on any part of the solution or have any questions?

Here are 5 related questions to expand on this concept:

1. How can you prove that a sequence diverges using limits?
2. What is the difference between absolute and conditional convergence for series?
3. How can L'Hôpital's Rule be applied to sequences?
4. Can we determine the speed of convergence for the sequence $a_n$?
5. How does the behavior of a sequence relate to the properties of its corresponding series?

Tip: When simplifying sequences, factor as much as possible to clearly see terms that approach zero as $n \to \infty$.