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

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}} ).

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

Compare the limit:
 The limit is ( \lim_{{n \to \infty}} \frac{n^2}{n^2 + 1} = 1 ).

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.

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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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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