General Contents
Detailed Contents
Index
Programmed tutorial: Sequence and L'Hopital's Rule
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If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
We wish to determine the convergence or divergence of the sequence given by
General Contents
Detailed Contents
Index
First, plot the terms of this series on paper.
Then check your graph by clicking on “Next”.
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Does it
look
like the sequence is converging?
Yes.
What appears to be the approximate limit?
The limit appears to be approximately 0.
Can we find the limit of
a
n
via the properties of a related function of x, where x is a real number?
Yes.
Set that up.
Is this a special case?
Yes
How?
The numerator and the denominator separately approach infinity.
What special technique can we use here?
L’Hopital’s Rule.
Describe
L’Hopital’s
Rule.
We find the limit by differentiating the numerator and the denominator separately, taking the ratio, and evaluating the limit of the resulting expression. (We do not use the Quotient Rule).
Set that up using the
notation.
Do the differentiation.
Can we find the limit now?
No.
Why?
We still have the numerator and denominator separately approaching infinity.
What can we do?
We can apply
L’Hopital’s
Rule again.
Do that.
Is the new numerator infinite now?
No.
Is the new denominator infinite now?
Yes.
Do we have to apply
L’Hopital’s
Rule again?
No.
What is the value of the limit?
The limit is 0.
What can we conclude here?
The sequence
converges to 0.
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