(a) Find at least ten partial sums of the series. (Round your answers to five decimal places.) $\displaystyle\sum_{n=1}^\infty \cos(9n).$

$n$	$s_n$
1	$\boxed{?}$
2	$\boxed{?}$
3	$\boxed{?}$
4	$\boxed{?}$
5	$\boxed{?}$
6	$\boxed{?}$
7	$\boxed{?}$
8	$\boxed{?}$
9	$\boxed{?}$
10	$\boxed{?}$

(b) Does it appear that the series is convergent or divergent? If it is convergent, find the sum. (If the quantity diverges, enter DIVERGES.)

$\boxed{?}$

Neetesh Kumar | December 27, 2024

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This is the solution to Math 1c
Assignment: 11.2 Question Number 4
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Step-by-step solution:

Step 1: Analyze the series:

The series involves the general term:

$a_n = \cos(9n).$

The partial sum $S_n$ is defined as:

$S_n = \displaystyle\sum_{k=1}^n \cos(9k).$

To evaluate the behavior of $S_n$, we need to compute the first $10$ partial sums.

Step 2: Compute the first $10$ partial sums:

Using a calculator or computational tool, evaluate:

1. For $n = 1$:

   $S_1 = \cos(9 \cdot 1) = \cos(9) \approx -0.911130.$

2. For $n = 2$:

   $S_2 = S_1 + \cos(9 \cdot 2) = -0.911130 + \cos(18) \approx -0.91113 + 0.6603167= -0.250813.$

3. For $n = 3$:

   $S_3 = S_2 + \cos(9 \cdot 3) = -0.250813 + \cos(27) \approx -0.250813 - 0.29213880 = -0.5429518.$

4. For $n = 4$:

   $S_4 = S_3 + \cos(9 \cdot 4) = -0.5429518 + \cos(36) \approx -0.5429518 - 0.12796 = -0.67091.$

5. For $n = 5$:

   $S_5 = S_4 + \cos(9 \cdot 5) = -0.67091 + \cos(45) \approx -0.67091 + 0.52532 = -0.14559.$

6. For $n = 6$:

   $S_6 = S_5 + \cos(9 \cdot 6) = -0.14559 + \cos(54) \approx -0.14559 - 0.82930 = -0.9749.$

7. For $n = 7$:

   $S_7 = S_6 + \cos(9 \cdot 7) = -0.9749 + \cos(63) \approx -0.9749 + 0.98589 = 0.01099.$

8. For $n = 8$:

   $S_8 = S_7 + \cos(9 \cdot 8) = 0.01099 + \cos(72) \approx 0.01099 - 0.967250 = -0.95625.$

9. For $n = 9$:

   $S_9 = S_8 + \cos(9 \cdot 9) = -0.95625 + \cos(81) \approx -0.95625 + 0.776685 = -0.17957.$

10. For $n = 10$:

    $S_{10} = S_9 + \cos(9 \cdot 10) = -0.17957 + \cos(90) \approx -0.17957 - 0.44807 = -0.62764.$

Step 3: Populate the table:

$n$	$S_n$
1	$-0.911130$
2	$-0.250813$
3	$-0.542952$
4	$-0.67091$
5	$-0.14559$
6	$-0.9749$
7	$0.01099$
8	$-0.95625$
9	$-0.17957$
10	$-0.62764$

Step 4: Convergence or divergence:

The partial sums $S_n$ do not settle to a fixed value as $n$ increases. Instead, the values oscillate due to the periodic nature of the cosine function. Therefore, the series diverges.

Final Answer:

(a)

$n$	$s_n$
1	$\boxed{-0.911130}$
2	$\boxed{-0.250813}$
3	$\boxed{-0.542952}$
4	$\boxed{-0.67091}$
5	$\boxed{-0.14559}$
6	$\boxed{-0.9749}$
7	$\boxed{0.01099}$
8	$\boxed{-0.95625}$
9	$\boxed{-0.17957}$
10	$\boxed{-0.62764}$

(b)

• The series $\boxed{\text{diverges}}$

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