How do you use the limit comparison test to determine if #Sigma (2n^2-1)/(3n^5+2n+1)# from #[1,oo)# is convergent or divergent?

Answer 1

The series:

#sum_(n=1)^oo (2n^2-1)/(3n^5+2n+1)#

is convergent.

Given:

#a_n = (2n^2-1)/(3n^5+2n+1)#
we have to find a convergent series #sum_(n=1)^oo b_n# such that #a_n < b_n#

We can now consider that:

#2n^2-1 < 2n^2#

and

# (3n^5+2n+1) > 3n^5#

so that:

#(2n^2-1)/(3n^5+2n+1) < (2n^2)/(3n^5)#
#(2n^2-1)/(3n^5+2n+1) < 1/n^3#
The series #sum_(n=1)^oo 1/n^3# is known to be convergent based on the p-series test, therefore also the series:
#sum_(n=1)^oo (2n^2-1)/(3n^5+2n+1)#

is convergent.

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Answer 2

To use the limit comparison test on the series ( \sum \frac{2n^2-1}{3n^5+2n+1} ) from ( n = 1 ) to ( \infty ):

  1. Select a simpler series that you know the convergence of.

  2. Compute the limit: [ \lim_{{n \to \infty}} \frac{{\frac{{2n^2 - 1}}{{3n^5 + 2n + 1}}}}{{\frac{1}{{n^p}}}} ] where ( p ) is the degree of the highest power in the denominator of the original series.

  3. If the limit is a finite positive number, then both series either converge or diverge. If it's zero or infinite, the behavior of both series may differ.

  4. Choose a simpler series wisely. Common choices include ( \sum \frac{1}{n^p} ) or ( \sum \frac{1}{n} ) depending on the degree of the polynomials in the original series.

  5. Simplify the limit as much as possible.

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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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