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

Answer 1
Let #a_n=({n^2+1]/{2n^2+1})^n#.

By Root Test,

#lim_{n to infty}root[n]{|a_n|}=lim_{n to infty}root[n]{|({n^2+1}/{2n^2+1})^n|}#

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

#=lim_{n to infty}{n^2+1}/{2n^2+1}#

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

by dividing by #n^2#,
#=lim_{n to infty}{1+1/n^2}/{2+1/n^2}={1+0}/{2+0}=1/2<1#

Hence, the series converges.

I hope that this was helpful.

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Answer 2

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

  1. Take the limit as (n) approaches infinity of the nth root of the absolute value of the nth term of the series.
  2. 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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Answer from HIX Tutor

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.

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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