# How do you test for convergence for #sum((-1)^n)*(sqrt(n))*(sin(1/n))# for n is 1 to infinity?

The series converges conditionally.

Using the chain rule, we get :

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The series is convergent.

We know that

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To test for convergence of the series (\sum_{n=1}^{\infty} (-1)^n \sqrt{n} \sin\left(\frac{1}{n}\right)), we can use the Alternating Series Test. The Alternating Series Test states that if a series has terms that alternate in sign and decrease in magnitude, and the limit of the absolute value of the terms approaches zero as (n) approaches infinity, then the series converges.

In this series, the terms alternate in sign due to the factor ((-1)^n). The term (\sqrt{n} \sin\left(\frac{1}{n}\right)) decreases in magnitude as (n) increases because both (\sqrt{n}) and (\sin\left(\frac{1}{n}\right)) decrease as (n) increases.

To confirm the limit of the absolute value of the terms approaches zero, we can analyze the behavior of the term (\sqrt{n} \sin\left(\frac{1}{n}\right)) as (n) approaches infinity. As (n) goes to infinity, (\frac{1}{n}) approaches zero, and therefore (\sin\left(\frac{1}{n}\right)) approaches zero. Additionally, (\sqrt{n}) also approaches infinity. Therefore, the product (\sqrt{n} \sin\left(\frac{1}{n}\right)) approaches zero as (n) approaches infinity.

Since the conditions of the Alternating Series Test are met (terms alternate in sign, decrease in magnitude, and limit of absolute value of terms approaches zero), we can conclude that the series (\sum_{n=1}^{\infty} (-1)^n \sqrt{n} \sin\left(\frac{1}{n}\right)) 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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