Math Problem Statement

Find at least ten partial sums of the series. (Round your answers to five decimal places.)

∞

cos(6n)

n = 1

    n            sn        

1   2   3   4   5   6   7   8   9   10   Graph both the sequence of terms and the sequence of partial sums on the same screen.

The x y-coordinate plane is given. There are 20 points on the graph, 10 points are colored blue and 10 points are colored black.

The approximate location of the 10 blue points are: 

(1, 0.9602)   (2, 0.8439)   (3, 0.6603)   (4, 0.4242)   (5, 0.1543)   (6, −0.1280)   (7, −0.4000)   (8, −0.6401)   (9, −0.8293)   (10, −0.9524)

.

The approximate location of the 10 black points are: 

(1, 0.9602)   (2, 1.8040)   (3, 2.4643)   (4, 2.8885)   (5, 3.0428)   (6, 2.9148)   (7, 2.5148)   (8, 1.8747)   (9, 1.0454)   (10, 0.0930)

.   The x y-coordinate plane is given. There are 20 points on the graph, 10 points are colored blue and 10 points are colored black.

The approximate location of the 10 blue points are: 

(1, −0.9602)   (2, 0.8439)   (3, −0.6603)   (4, 0.4242)   (5, −0.1543)   (6, −0.1280)   (7, 0.4000)   (8, −0.6401)   (9, 0.8293)   (10, −0.9524)

.

The approximate location of the 10 black points are: 

(1, −0.9602)   (2, 1.8040)   (3, −2.4643)   (4, 2.8885)   (5, −3.0428)   (6, 2.9148)   (7, −2.5148)   (8, 1.8747)   (9, −1.0454)   (10, 0.0930)

.   The x y-coordinate plane is given. There are 20 points on the graph, 10 points are colored blue and 10 points are colored black.

The approximate location of the 10 blue points are: 

(1, 0.9602)   (2, −0.8439)   (3, 0.6603)   (4, −0.4242)   (5, 0.1543)   (6, 0.1280)   (7, −0.4000)   (8, 0.6401)   (9, −0.8293)   (10, 0.9524)

.

The approximate location of the 10 black points are: 

(1, 0.9602)   (2, −1.8040)   (3, 2.4643)   (4, −2.8885)   (5, 3.0428)   (6, −2.9148)   (7, 2.5148)   (8, −1.8747)   (9, 1.0454)   (10, −0.0930)

.   The x y-coordinate plane is given. There are 20 points on the graph, 10 points are colored blue and 10 points are colored black.

The approximate location of the 10 blue points are: 

(1, −0.9602)   (2, −0.8439)   (3, −0.6603)   (4, −0.4242)   (5, −0.1543)   (6, 0.1280)   (7, 0.4000)   (8, 0.6401)   (9, 0.8293)   (10, 0.9524)

.

The approximate location of the 10 black points are: 

(1, −0.9602)   (2, −1.8040)   (3, −2.4643)   (4, −2.8885)   (5, −3.0428)   (6, −2.9148)   (7, −2.5148)   (8, −1.8747)   (9, −1.0454)   (10, −0.0930)

.

Solution

I have displayed the first ten terms and their partial sums in the table, as well as plotted the sequence of terms and partial sums. This visual representation helps show how the series behaves as more terms are added.

Do you have any questions, or would you like additional details?

Here are five related questions to further explore the topic:

  1. What is the behavior of the partial sums as nn approaches infinity?
  2. How does the sequence of terms an=cos⁡(6n)/na_n = \cos(6n) / n contribute to the convergence or divergence of the series?
  3. What are the implications of the oscillatory nature of cos⁡(6n)\cos(6n) in this series?
  4. Could this series be approximated by any known convergence tests?
  5. How would the series change if we used sin⁡(6n)\sin(6n) instead of cos⁡(6n)\cos(6n)?

Tip: When analyzing infinite series, comparing the terms' growth rate to known convergent or divergent series can be a useful approach to determining convergence behavior.

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Math Problem Analysis

Mathematical Concepts

Infinite Series
Partial Sums
Trigonometric Functions
Cosine Series

Formulas

Partial Sum Formula for Series S_n = Σ cos(6n)/n from n = 1 to N

Theorems

Convergence Test for Infinite Series
Oscillatory Behavior of Trigonometric Series

Suitable Grade Level

Grades 10-12

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