How do you use the integral test to determine if #Sigma n/(n^4+1)# from #[1,oo)# is convergent or divergent?
If the integral diverges, so does the series.
So, we want to test the convergence of the integral:
Since this integral converges, so does the given series via the integral test.
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To use the integral test to determine the convergence or divergence of the series (\sum_{n=1}^{\infty} \frac{n}{n^4 + 1}), you need to follow these steps:

Evaluate the corresponding improper integral: [ \int_{1}^{\infty} \frac{x}{x^4 + 1} , dx ]

Determine if the integral converges or diverges.

If the integral converges, then the series (\sum_{n=1}^{\infty} \frac{n}{n^4 + 1}) converges. If the integral diverges, then the series also diverges.
So, to determine convergence or divergence, evaluate the integral (\int_{1}^{\infty} \frac{x}{x^4 + 1} , dx) and ascertain its convergence or divergence.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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