# How do you determine if the improper integral converges or diverges #int dx/((3x-2)^6) # from 2 to infinity?

The integral is convergent and:

By definition:

We have that:

So:

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To determine if the improper integral ( \int_{2}^{\infty} \frac{1}{(3x - 2)^6} , dx ) converges or diverges, we need to analyze its behavior as ( x ) approaches infinity.

We'll use the limit comparison test with a known integral. Let's compare it to the integral ( \int_{2}^{\infty} \frac{1}{x^6} , dx ).

[ \lim_{x \to \infty} \frac{\frac{1}{(3x - 2)^6}}{\frac{1}{x^6}} ]

By simplifying, we get:

[ \lim_{x \to \infty} \frac{x^6}{(3x - 2)^6} ]

[ = \lim_{x \to \infty} \left(\frac{x}{3x - 2}\right)^6 ]

As ( x ) approaches infinity, the expression ( \frac{x}{3x - 2} ) approaches ( \frac{1}{3} ).

Thus, the limit becomes ( \left(\frac{1}{3}\right)^6 = \frac{1}{729} ).

Since the limit is a finite non-zero number, and the integral ( \int_{2}^{\infty} \frac{1}{x^6} , dx ) converges (it's a p-series with ( p > 1 )), by the limit comparison test, the given integral ( \int_{2}^{\infty} \frac{1}{(3x - 2)^6} , dx ) 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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