# How do you apply the ratio test to determine if #sum_(n=2)^oo 10^n/(lnn)^n# is convergent to divergent?

The Ratio Test can be used to show that this series converges.

All this implies that

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To apply the ratio test to determine if the series ∑(n=2)^∞ 10^n / (lnn)^n is convergent or divergent, we first compute the limit of the absolute value of the ratio of successive terms as n approaches infinity.

Let's denote the nth term of the series as a_n = 10^n / (lnn)^n. Then, we calculate the ratio of consecutive terms:

r_n = |a_(n+1) / a_n| = |(10^(n+1) / (ln(n+1))^(n+1)) / (10^n / (lnn)^n)|

Simplify the expression:

r_n = |10^(n+1) / (ln(n+1))^(n+1) * (lnn)^n / 10^n|

= |10 / (ln(n+1) / lnn)^(n+1)|

= |10 * (lnn / ln(n+1))^(n+1)|

Now, we take the limit as n approaches infinity:

lim (n→∞) |10 * (lnn / ln(n+1))^(n+1)|

Since (lnn / ln(n+1))^(n+1) approaches 1 as n approaches infinity, the limit simplifies to:

lim (n→∞) |10| = 10

Since the limit of the absolute value of the ratio is greater than 1, by the ratio test, the series ∑(n=2)^∞ 10^n / (lnn)^n is divergent.

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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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