The image describes an infinite sequence with terms $\frac{1}{2}, \frac{2}{4}, \frac{3}{8}, \frac{4}{16}, \dots$ and asks to find its general term ($a_n$) and determine whether the sequence converges or diverges.

Step 1: Identify the general form of the sequence.

Looking at the terms, the numerators follow the pattern $1, 2, 3, 4, \dots$, which indicates the sequence of natural numbers $n$. The denominators are powers of 2: $2, 4, 8, 16, \dots$, which correspond to $2^n$.

Thus, the general term $a_n$ of the sequence can be expressed as: $a_n = \frac{n}{2^n}$

Step 2: Determine if the sequence converges or diverges.

To determine if the sequence converges or diverges, we need to check the limit of $a_n$ as $n \to \infty$.

$\lim_{n \to \infty} \frac{n}{2^n}$

Since the denominator grows exponentially faster than the numerator, this limit equals 0. Therefore, the sequence converges to 0.

Conclusion:

• The general formula for the sequence is $a_n = \frac{n}{2^n}$.
• The sequence converges to 0.

Let me know if you'd like further details or have any questions! Here are some related questions to explore:

1. How do you prove that $\frac{n}{2^n} \to 0$ as $n \to \infty$?
2. Can this sequence be considered a geometric series or something else?
3. What is the difference between a converging sequence and a divergent sequence?
4. How would the sequence behave if the denominator were $3^n$ instead of $2^n$?
5. How can you find the sum of the series if it converges?

Tip: When analyzing the behavior of sequences, always consider the rate of growth of both the numerator and the denominator.