https://file.aiquickdraw.com/mathbot/file/1726395450808cu0g6iyi.jpg

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.

Sebuah barisan tak hingga sbb: 1/2, 2/4, 3/8, 4/16, ... Tentukan rumusan polanya (an) setelah ditemukan ujilah barisan ini konvergen atau divergen.

This problem explores the infinite sequence 1/2, 2/4, 3/8, 4/16, ... and determines its general formula (an = n / 2^n). Learn how to calculate the limit and establish that the sequence converges to 0. Suitable for high school students studying sequences and series.

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