Patterns in Integrals



Lab 2: Series

Seattle Pacific University, MAT 1236, Calculus III

Objectives

➢ This lab helps you to explore and understand

❑ the convergence of a series

❑ some examples of standard series

Due Date: Friday 04/14 2:01 p.m.

❑ Do not wait until the last minute to finish the lab. You never know what technical problems you may encounter (no papers in the printer, electricity is out, your best friend call and talk for 4 hours, alien attack…etc).

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|Name 1: Key |

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|Name 2: |

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|Date: |

Lab Exercises

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|Given a series [pic], for [pic], we define its partial sum sequence [pic]as |

|[pic] |

|That is, [pic] is the sum of the first [pic]terms of the series. |

1. Given the series

[pic]

Use the following statement to define the partial sum sequence [pic] in a Maple worksheet.

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|> S:=n->sum(6/k^2,k=1..n); |

|[pic] |

To find the value of [pic], we simply type:

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|> S(3); |

|[pic] |

Or,

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|> evalf(S(3)); |

|[pic] |

(a) Use Maple to compute the values of the following terms of the partial sum sequence [pic]. Round your answers to 9 decimal places.

|[pic] |[pic] |

|1 |6.000000000 |

|2 |7.500000000 |

|3 |8.166666667 |

|5 |8.781666667 |

|100 |9.809903401 |

|10000 |9.869004431 |

|100000 |9.869544401 |

|200000 |9.869574401 |

(b) Do you think the sequence converges?

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|Yes. |

If so, what value it probably converges to? (Hint: Take the square root of[pic].)

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|[pic] |

(c) What do you think the value of the series [pic]should equals to?

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|[pic] |

Remark [pic] is an example of a p-series (Section 12.3).

2. Given the series

[pic]

(a) Use Maple to compute the values of the following terms of the partial sum sequence [pic].

|[pic] |[pic] |

|1 |6 |

|2 |30 |

|3 |84 |

|10 |2310 |

|20 |17220 |

|30 |56730 |

|40 |132840 |

|50 |257550 |

(b) Do you think the sequence converges?

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|No. |

If so, what value it probably converges to? (Ignore this if the sequence diverges.)

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|(Ignore) |

(c) What can you say about the series [pic]?

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|[pic] diverges. |

Remark [pic] is an example of a p-series (Section 12.3).

3. Given the series

[pic]

(a) Use Maple to compute the values of the following terms of the partial sum sequence [pic]. Round your answers to 9 decimal places.

|[pic] |[pic] |

|10 |1.998046875 |

|15 |1.999938965 |

|24 |1.999999881 |

|26 |1.999999970 |

|28 |1.999999993 |

|30 |1.999999998 |

|32 |2.000000000 |

|40 |2.000000000 |

(b) Do you think the sequence converges?

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|Yes. |

If so, what value it probably converges to? (Ignore this if the sequence diverges.)

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|[pic] |

(c) What can you say about the series [pic]?

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|[pic] |

Remarks [pic] is an example of a Geometric series (Section 12.2).

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