How to choose the Bn for limit comparison test?

If the An is #(e^(1/n))/n#
how would you determine what bn to use to compare with this?

Answer 1

Note that #e^{1/n}>1# for all integers #n>0#. Therefore, we expect that #sum_{n=1}^{infty}e^{1/n}/n# will diverge. Try comparing it to the divergent harmonic series #sum_{n=1}^{infty}1/n# to show this with the limit comparison test (so use #b_{n}=1/n#).

Let #a_{n}=e^{1/n}/n# and #b_{n}=1/n#, noting that #a_{n} > b_{n} > 0# for all integers #n>0#.
Now compute #lim_{n->infty}a_{n}/b_{n}#. We are"hoping" it is a positive number and not #infty#, which will allow us to say that #sum_{n=1}^{infty}e^{1/n}/n# diverges by the Limit Comparison Test since we know that the harmonic series #sum_{n=1}^{infty}1/n# diverges.
But clearly, #lim_{n->infty}a_{n}/b_{n}=lim_{n->infty}e^{1/n}=1#, a positive number (and not #infty#). We are done.
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Answer 2

To choose ( B_n ) for the limit comparison test, follow these steps:

  1. Select a series ( \sum a_n ) that you suspect behaves similarly to the series ( \sum b_n ).
  2. Choose ( B_n ) such that ( 0 < B_n < a_n ) for all ( n ) beyond some point.
  3. Compute the limit ( \lim_{n \to \infty} \frac{a_n}{B_n} ).
  4. If the limit is a positive finite number, then ( \sum a_n ) and ( \sum B_n ) either both converge or both diverge.
  5. If the limit is zero or infinity, consider choosing a different ( B_n ) and repeat the process until the limit meets the conditions outlined in step 4.
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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