# How do you test for convergence of #Sigma (-1)^n/sqrt(lnn)# from #n=[3,oo)#?

The series converges by the alternating series test.

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To test for the convergence of the series ( \sum_{n=3}^{\infty} \frac{(-1)^n}{\sqrt{\ln n}} ), you can use the alternating series test. This test states that if a series is alternating, meaning that its terms alternate in sign, and the absolute value of the terms decreases as ( n ) increases, and the terms approach zero as ( n ) approaches infinity, then the series converges.

In this series, the terms alternate in sign and approach zero as ( n ) approaches infinity. To check if the terms decrease in absolute value, you can observe the behavior of the function ( \frac{1}{\sqrt{\ln n}} ) as ( n ) increases.

Since ( \ln n ) increases as ( n ) increases, ( \sqrt{\ln n} ) also increases. Therefore, ( \frac{1}{\sqrt{\ln n}} ) decreases as ( n ) increases.

Because the series meets all the conditions of the alternating series test, it 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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