# Using the integral test, how do you show whether #sum1/[(n^2)+4)# diverges or converges from n=1 to infinity?

But now on to the integral test as requested:

In other words, the improper integral converges.

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To determine whether the series ( \sum_{n=1}^{\infty} \frac{1}{n^2 + 4} ) converges or diverges, we can use the integral test.

- Define the function ( f(x) = \frac{1}{x^2 + 4} ).
- Verify that ( f(x) ) is positive, continuous, and decreasing for ( x \geq 1 ).
- Integrate ( f(x) ) from ( 1 ) to ( \infty ).

[ \int_{1}^{\infty} \frac{1}{x^2 + 4} , dx ]

- Evaluate the integral and determine whether the result is finite or infinite.

If the integral converges, then the series ( \sum_{n=1}^{\infty} \frac{1}{n^2 + 4} ) converges. If the integral diverges, then the series 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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