# How do you apply the ratio test to determine if #Sigma 1/sqrtn# from #n=[1,oo)# is convergent to divergent?

The series diverges, but this Ratio Test cannot determine this.

The problem is this limit comes out to 1, meaning the Ratio Test will not tell you the convergence or divergence of this series. You must use another test, such as the Integral Test or the Comparison Test (shown here ).

In any case, the series diverges.

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To apply the ratio test to the series Σ(1/√n) from n=1 to infinity, you need to compute the limit of the ratio of successive terms as n approaches infinity.

First, find the general term of the series, which is ( a_n = \frac{1}{\sqrt{n}} ).

Then, compute the ratio of successive terms: [ \frac{a_{n+1}}{a_n} = \frac{\frac{1}{\sqrt{n+1}}}{\frac{1}{\sqrt{n}}} ]

Simplify the ratio: [ \frac{\sqrt{n}}{\sqrt{n+1}} ]

To evaluate the limit as n approaches infinity: [ \lim_{n \to \infty} \frac{\sqrt{n}}{\sqrt{n+1}} = 1 ]

Since the limit is equal to 1, the ratio test is inconclusive. Therefore, you cannot determine convergence or divergence solely using the ratio test for this series.

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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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