# How do you use the direct Comparison test on the infinite series #sum_(n=2)^oon^3/(n^4-1)# ?

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To use the Direct Comparison Test on the infinite series ( \sum_{n=2}^\infty \frac{n^3}{n^4 - 1} ), we need to find a series with terms that are easier to evaluate and that are always greater than or equal to the terms of the given series.

Notice that for all ( n \geq 2 ), [ n^4 - 1 > n^4 - n^4 = 0 ] Thus, [ \frac{n^3}{n^4 - 1} < \frac{n^3}{0} ] is undefined. However, let's look at a slightly modified series: ( \sum_{n=1}^\infty \frac{n^3}{n^4} ).

For this series, we have: [ \frac{n^3}{n^4} = \frac{1}{n} ]

The series ( \sum_{n=1}^\infty \frac{1}{n} ) is a known divergent series, the harmonic series.

Now, since ( \frac{n^3}{n^4 - 1} ) is less than ( \frac{n^3}{n^4} ) for all ( n \geq 2 ), and ( \sum_{n=1}^\infty \frac{1}{n} ) diverges, we can conclude by the Direct Comparison Test that ( \sum_{n=2}^\infty \frac{n^3}{n^4 - 1} ) also diverges.

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To use the Direct Comparison Test on the infinite series ( \sum_{n=2}^\infty \frac{n^3}{n^4 - 1} ), we need to compare it with a known series whose convergence behavior is understood.

First, we observe that for large values of ( n ), ( n^4 - 1 ) is approximately ( n^4 ). Thus, we can compare the given series with the series ( \sum_{n=2}^\infty \frac{n^3}{n^4} = \sum_{n=2}^\infty \frac{1}{n} ).

The series ( \sum_{n=2}^\infty \frac{1}{n} ) is a p-series with ( p = 1 ). It is a known result that this series diverges (Harmonic Series).

Since ( \frac{n^3}{n^4 - 1} < \frac{1}{n} ) for all ( n \geq 2 ), and the series ( \sum_{n=2}^\infty \frac{1}{n} ) diverges, by the Direct Comparison Test, the series ( \sum_{n=2}^\infty \frac{n^3}{n^4 - 1} ) also diverges.

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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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