# How do you use the limit comparison test on the series #sum_(n=1)^oo(n+1)/(n*sqrt(n))# ?

The series:

is divergent.

Note that:

it follows that:

and as the series:

is also divergent.

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To use the limit comparison test on the series ( \sum_{n=1}^\infty \frac{n+1}{n\sqrt{n}} ), we compare it with a known convergent or divergent series.

Let's choose the series ( \sum_{n=1}^\infty \frac{1}{\sqrt{n}} ), which is a known convergent series.

Then, we take the limit as ( n ) approaches infinity of the ratio of the terms of the two series:

[ \lim_{n \to \infty} \frac{\frac{n+1}{n\sqrt{n}}}{\frac{1}{\sqrt{n}}} ]

Simplify the expression:

[ \lim_{n \to \infty} \frac{n+1}{n} = 1 ]

Since the limit is a finite positive number, and the series ( \sum_{n=1}^\infty \frac{1}{\sqrt{n}} ) is convergent, by the limit comparison test, the series ( \sum_{n=1}^\infty \frac{n+1}{n\sqrt{n}} ) is also convergent.

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To use the limit comparison test on the series (\sum_{n=1}^{\infty} \frac{n+1}{n\sqrt{n}}), you would choose another series whose convergence behavior is known and compare it to the given series.

In this case, we can consider the series (\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}), which is a well-known series (the p-series with (p = \frac{1}{2})).

Now, we take the limit as (n) approaches infinity of the ratio of the terms of the two series:

[\lim_{n \to \infty} \frac{(n+1)/(n\sqrt{n})}{1/\sqrt{n}}]

Simplify this expression to find the limit:

[\lim_{n \to \infty} \frac{n+1}{n\sqrt{n}} \times \frac{\sqrt{n}}{1}]

[= \lim_{n \to \infty} \frac{n+1}{n}]

[= 1]

Since this limit is a finite, positive number, and the series (\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}}) converges (as a p-series with (p = \frac{1}{2})), then by the limit comparison test, the series (\sum_{n=1}^{\infty} \frac{n+1}{n\sqrt{n}}) 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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