# How do you use the limit comparison test on the series #sum_(n=1)^oon/(2n^3+1)# ?

Let us look at some details.

#lim_{n to infty}a_n/b_n=lim_{n to infty}n/{2n^3+1}cdotn^2/1 =lim_{n to infty}n^3/{2n^3+1}#

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To use the limit comparison test on the series (\sum_{n=1}^\infty \frac{n}{2n^3+1}), you need to compare it with a known convergent or divergent series. Choose a series (b_n) such that:

- (b_n > 0) for all (n),
- (\lim_{n \to \infty} \frac{a_n}{b_n}) exists and is a finite positive number.

Then, apply the limit comparison test:

- If (\sum_{n=1}^\infty b_n) converges, then (\sum_{n=1}^\infty a_n) also converges.
- If (\sum_{n=1}^\infty b_n) diverges, then (\sum_{n=1}^\infty a_n) also diverges.

In this case, we can choose (b_n = \frac{1}{n^2}). Now, we evaluate the limit:

[\lim_{n \to \infty} \frac{\frac{n}{2n^3+1}}{\frac{1}{n^2}} = \lim_{n \to \infty} \frac{n^3}{2n^3+1} = \frac{1}{2}]

Since the limit exists and is a finite positive number, and (\sum_{n=1}^\infty \frac{1}{n^2}) converges (by p-series test with (p = 2 > 1)), by the limit comparison test, the given series (\sum_{n=1}^\infty \frac{n}{2n^3+1}) also converges.

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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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