1
1. If a series [pic]diverges, what possible conclusion could we draw about the sequence [pic]?
The sequence [pic] converges to 0.
The sequence [pic] diverges.
The sequence [pic] converges to a non-zero value.
Which one of the following statements is most correct?
Conclusion (I) is the only possible conclusion for this situation.
Conclusion (II) is the only possible conclusion for this situation.
Conclusion (III) is the only possible conclusion for this situation.
Conclusions (I) and (II) are the only possible conclusions for this situation.
Conclusions (II) and (III) are the only possible conclusions for this situation.
Any one of the three conclusions could be true for this situation.
2. Provide an example of a sequence that is both decreasing and bounded below. Respond by expressing a general term of the sequence, an, or by listing the first six terms of the sequence. Explain how you know the sequence is decreasing and indicate a value for the lower bound.
(a) Example sequence or general term:
(b) Explain how you know it is decreasing:
(c) State a value for the lower bound:
3. Calculate the 5th partial sum, s5, to estimate the value of [pic]. Express your response as a common fraction reduced to lowest terms.
4. Does the series [pic] converge or diverge? Provide complete justification for your response.
5. What can you conclude about the series [pic]if you know that the sequence [pic] converges to 0? Explain your response.
BONUS!
For the series shown in (3), [pic], what is the smallest value of n that assures us that [pic]? Explain how you determined your response.
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