Does #a_n=(2+n+(n^3))/sqrt(2+(n^2)+(n^8)) #converge? If so what is the limit?

Answer 1

The sequence converges and the limit is #0#.

When one has to work with sequences that can be reduced to a quotient of sums (and some variations of them, like in this case that embeds a square root), it's always a good idea to work as follows: at first study the numerator and find the term of the sum that grows faster and factor it out; then do the same for the denominator; finally, compare the two terms and get the behaviour of the limit.

In the specific case, we have

So the limit becomes #lim_{n to +infty} a_n=lim_{n to +infty} [n^3(2/n^3+1/n^2+1)]/[n^4 sqrt(2/n^8+1/n^6+1)]=lim_{n to +infty} [n^3 * 1]/[n^4 * 1]=lim_{n to +infty} 1/n=0#.
This shows that the sequence converges and the limit is #0#.
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Answer 2

To determine if the sequence ( a_n = \frac{2 + n + n^3}{\sqrt{2 + n^2 + n^8}} ) converges, we'll analyze its behavior as ( n ) approaches infinity.

The highest power terms in the numerator and denominator are ( n^3 ) and ( n^8 ), respectively. Since the degree of the denominator is greater than the degree of the numerator, we can simplify the expression by dividing both numerator and denominator by ( n^4 ):

[ a_n = \frac{\frac{2}{n^4} + \frac{n}{n^4} + 1}{\sqrt{\frac{2}{n^4} + 1 + n^4}} ]

As ( n ) approaches infinity, all terms with ( n ) in the numerator tend to 0. We're then left with:

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

Since the limit exists and is finite, the sequence converges, and its limit is 1.

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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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