How do you use the limit comparison test to determine if #Sigma n/(n^2+1)# from #[1,oo)# is convergent or divergent?

Answer 1

See below.

#n/(n^2+n) < n/(n^2+1) < n/n^2# or
#1/(n+1) < n/(n^2+1) < 1/n#
Here #sum_(n=1)^oo1/(n+1) le sum_(n=1)^oon/(n^2+1)le sum_(n=1)^oo1/n#

but

#1+sum_(n=1)^oo1/(n+1) =sum_(n=1)^oo1/n =oo#
so #sum_(n=1)^oon/(n^2+1)# is divergent
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Answer 2
  1. Determine the comparison function:

    • Choose a series that is easier to work with or known to converge/diverge.
    • For ( \sum \frac{n}{n^2 + 1} ), consider ( \sum \frac{1}{n} ) which is a known series.
  2. Take the limit of the ratio:

    • Consider ( \lim_{{n \to \infty}} \frac{a_n}{b_n} ), where ( a_n = \frac{n}{n^2 + 1} ) and ( b_n = \frac{1}{n} ).
    • ( \lim_{{n \to \infty}} \frac{a_n}{b_n} = \lim_{{n \to \infty}} \frac{\frac{n}{n^2 + 1}}{\frac{1}{n}} ).
  3. Simplify the limit:

    • Simplify the expression: ( \lim_{{n \to \infty}} \frac{n^2}{n^2 + 1} ).
  4. Compare the limit:

    • The limit is ( \lim_{{n \to \infty}} \frac{n^2}{n^2 + 1} = 1 ).
  5. Conclude using the Limit Comparison Test:

    • Since ( \lim_{{n \to \infty}} \frac{a_n}{b_n} = 1 ), and ( \sum \frac{1}{n} ) is a known series (Harmonic series which diverges), then ( \sum \frac{n}{n^2 + 1} ) behaves like ( \sum \frac{1}{n} ).
    • By the Limit Comparison Test, if the comparison series diverges, then ( \sum \frac{n}{n^2 + 1} ) also diverges.
  6. Conclusion:

    • ( \sum \frac{n}{n^2 + 1} ) is divergent since it behaves like the divergent Harmonic series ( \sum \frac{1}{n} ).
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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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