# How do you use the Nth term test for divergence on an alternating series?

If an alternating series has the form

then the series diverges if

Example

Let us look at the alternating series below.

By Nth Terms (Divergence) Test,

Hence, the series diverges.

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To use the Nth term test for divergence on an alternating series, follow these steps:

- Evaluate the limit as (n) approaches infinity of the absolute value of the (n)th term of the series.
- If the limit is not zero, the series diverges.
- If the limit is zero, the test is inconclusive, and other tests or methods may need to be used to determine convergence or divergence.

Mathematically, the Nth term test for divergence on an alternating series can be stated as follows:

Given an alternating series (\sum_{n=1}^{\infty} (-1)^{n+1} a_n), where (a_n) is the (n)th term of the series, the series diverges if:

[\lim_{n \to \infty} |a_n| \neq 0]

Otherwise, if the limit is zero:

[\lim_{n \to \infty} |a_n| = 0]

then the test is inconclusive, and other methods or tests need to be employed to determine convergence or divergence.

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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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