# How do you use the Root Test on the series #sum_(n=1)^oo((n^2+1)/(2n^2+1))^(n)# ?

By Root Test,

by cancelling out the nth-root and the nth-power,

(Note: the absolute value is not necessary since it is already positive.)

Hence, the series converges.

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To use the Root Test on the series (\sum_{n=1}^{\infty} \left(\frac{n^2+1}{2n^2+1}\right)^n):

- Take the limit as (n) approaches infinity of the nth root of the absolute value of the nth term of the series.
- If the limit is less than 1, the series converges. If it's greater than 1 or the limit does not exist, the series diverges.

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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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