# How do you use the limit comparison test for #sum (2x^4)/(x^5+10)# n=1 to #n=oo#?

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To use the limit comparison test for the series ( \sum \frac{2x^4}{x^5 + 10} ) as ( n ) approaches infinity, we need to compare it with a known series. Let's denote the given series as ( a_n = \frac{2x^4}{x^5 + 10} ).

First, we find a series ( b_n ) that is easier to evaluate but behaves similarly to ( a_n ). In this case, we can choose ( b_n = \frac{1}{x} ) as it is simpler and has similar terms in the denominator.

Next, we take the limit as ( n ) approaches infinity of the ratio ( \frac{a_n}{b_n} ):

[ \lim_{{n \to \infty}} \frac{a_n}{b_n} = \lim_{{n \to \infty}} \frac{\frac{2x^4}{x^5 + 10}}{\frac{1}{x}} = \lim_{{n \to \infty}} \frac{2x^5}{x^5 + 10} ]

Solving the limit, we get:

[ \lim_{{n \to \infty}} \frac{2x^5}{x^5 + 10} = \frac{2}{1} = 2 ]

Since the limit is a finite positive number, we conclude that the series ( \sum \frac{2x^4}{x^5 + 10} ) behaves similarly to the series ( \sum \frac{1}{x} ) as ( n ) approaches infinity.

Now, we can apply the limit comparison test. Since ( \sum \frac{1}{x} ) is a known divergent series (harmonic series), and ( \sum \frac{2x^4}{x^5 + 10} ) behaves similarly to it, we conclude that ( \sum \frac{2x^4}{x^5 + 10} ) diverges as well.

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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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