# What is the root test?

Given:

and:

then:

The root test states that given a series with positive terms:

then we have:

In fact, suppose that:

Now:

is convergent by direct comparison.

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The root test is a method used in calculus to determine the convergence or divergence of an infinite series. It states that if ( \lim_{n \to \infty} \sqrt[n]{|a_n|} ) exists and is less than 1, then the series ( \sum_{n=1}^{\infty} a_n ) converges absolutely. If the limit is greater than 1 or does not exist, the series diverges. If the limit equals 1, the root test is inconclusive.

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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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