# How do you use the integral test to determine the convergence or divergence of #Sigma 1/n^2# from #[1,oo)#?

The series

The integral test states that, given:

then a necessary and sufficient condition for the series above to converge is that the integral:

converges as well.

It's easy to see that for the series:

so that the series must be convergent as well.

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To use the integral test to determine the convergence or divergence of ( \sum_{n=1}^{\infty} \frac{1}{n^2} ) from ( n = 1 ) to ( n = \infty ), we first integrate the function ( f(x) = \frac{1}{x^2} ) from ( x = 1 ) to ( x = \infty ). If the integral converges, then the series ( \sum_{n=1}^{\infty} \frac{1}{n^2} ) converges. If the integral diverges, then the series ( \sum_{n=1}^{\infty} \frac{1}{n^2} ) also diverges.

The integral of ( f(x) = \frac{1}{x^2} ) from ( x = 1 ) to ( x = \infty ) is given by:

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

We evaluate this integral to determine its convergence or divergence. If the integral converges, then the series ( \sum_{n=1}^{\infty} \frac{1}{n^2} ) converges. If the integral diverges, then the series ( \sum_{n=1}^{\infty} \frac{1}{n^2} ) also diverges.

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To use the integral test to determine the convergence or divergence of the series Σ 1/n^2 from 1 to infinity:

- Consider the function f(x) = 1/x^2.
- Integrate f(x) from 1 to infinity. If the integral converges, then the series converges. If the integral diverges, then the series also diverges.
- Evaluate the integral ∫(1/x^2) dx from 1 to infinity.
- The integral ∫(1/x^2) dx = [-1/x] from 1 to infinity = [-1/infinity] - [-1/1] = 0 - (-1) = 1.
- Since the integral equals 1, it converges.
- Therefore, by the integral test, the series Σ 1/n^2 from 1 to infinity 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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