009C Sample Midterm 1, Problem 1
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Does the following sequence converge or diverge?
If the sequence converges, also find the limit of the sequence.
Be sure to jusify your answers!
Foundations: |
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L'Hôpital's Rule |
Suppose that and are both zero or both |
If is finite or |
then |
Solution:
Step 1: |
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First, notice that |
and |
Therefore, the limit has the form |
which means that we can use L'Hopital's Rule to calculate this limit. |
Step 2: |
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First, switch to the variable so that we have functions and |
can take derivatives. Thus, using L'Hopital's Rule, we have |
Final Answer: |
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The sequence converges. The limit of the sequence is |