https://file.aiquickdraw.com/mathbot/file/1729884560748vrz9h3d2.jpg

The series presented is:

\[
\sum_{k=1}^{\infty} (\cos(1.2))^k
\]

### Step 1: Understanding the series structure
The series can be written as:

\[
\sum_{k=1}^{\infty} r^k
\]

where \( r = \cos(1.2) \).

### Step 2: Evaluate \(\cos(1.2)\)
The cosine of 1.2 radians is approximately:

\[
\cos(1.2) \approx 0.3624
\]

Thus, the series becomes:

\[
\sum_{k=1}^{\infty} (0.3624)^k
\]

### Step 3: Test for convergence
This is a geometric series of the form:

\[
\sum_{k=1}^{\infty} r^k
\]

A geometric series converges if the absolute value of the ratio \( r \) is less than 1, i.e., \( |r| < 1 \).

Since \( |\cos(1.2)| \approx 0.3624 \), which is less than 1, the series converges.

### Step 4: Finding the sum of the series
For a geometric series that starts at \( k = 1 \), the sum is given by:

\[
S = \frac{r}{1 - r}
\]

Substituting \( r = 0.3624 \):

\[
S = \frac{0.3624}{1 - 0.3624} = \frac{0.3624}{0.6376} \approx 0.5684
\]

### Final Answer:
- The series **converges**.
- The value of the series is approximately \( 0.5684 \).

Would you like any further clarification or details on the steps?

### Related Questions:
1. What is the general formula for the sum of an infinite geometric series?
2. How can we generalize this process for any angle \( \theta \) in cosine functions?
3. What happens if \( |\cos(\theta)| \geq 1 \) for series convergence?
4. How do we handle geometric series when the ratio \( r \) is negative?
5. How do you determine the radius of convergence for other types of series?

### Tip:
For infinite geometric series, always check the absolute value of the ratio to confirm convergence.

Consider the series ∑(cos 1.2)^k from k=1 to infinity. Determine whether the series converges, and if it converges, determine its value.

Learn how to determine the convergence of the series ∑(cos 1.2)^k from k=1 to infinity and calculate its sum. This problem involves understanding geometric series and applying the sum formula to find the value of the convergent series.

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