# How do you use the integral test to determine the convergence or divergence of #1+1/4+1/9+1/16+1/25+...#?

Our first step is to write the series in series notation. Notice that:

Then:

Evaluating:

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To use the integral test to determine the convergence or divergence of the series 1 + 1/4 + 1/9 + 1/16 + 1/25 + ..., we need to consider the corresponding function ( f(x) = \frac{1}{x^2} ).

- First, note that ( f(x) ) is positive, continuous, and decreasing for ( x \geq 1 ).
- Then, we integrate ( f(x) ) from 1 to infinity. This gives us ( \int_{1}^{\infty} \frac{1}{x^2} , dx = 1 ).
- Since the integral converges, by the integral test, the series ( \sum_{n=1}^{\infty} \frac{1}{n^2} ) also converges.

Therefore, the series 1 + 1/4 + 1/9 + 1/16 + 1/25 + ... 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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