Convergence and Divergence
Which of the sequences {aₙ} i...

Question

Convergence and Divergence

Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.

aₙ = (n + 3) / (n² + 5n + 6)

Accepted Answer

Welcome back everyone. Does the sequence a n equals n + 4 divided by n2 + 7 n + 12 converge or diverge? If it converges, state its limit. For this problem, let's begin by applying the nth term test for convergence of sequences. We want to evaluate limit as n approaches infinity of a m, and in this case that will be limit as n approaches infinity of n + 4 divided by n2 + 7 n + 12. And essentially when an approaches infinity, both the numerator and the denominator approach infinity. So what we can do is simply divide both sides by n 2, and we're going to get n divided by n2 plus 4 divided by n2 in the denominator that's going to be n2 divided by n2 + 7 n divided by n2 plus 12 divided by n2. Let's go ahead and divide. We get limits and approaches infinity of 1 divided by n + 4 divided by n2 divided by 1 + 7 divided by n + 12 divided by n 2. So now the fractional terms will approach 0 because we have a constant divided by infinity, meaning we have two 0s in the numerator and two 0s in the denominator, and we basically get 0 divided by 1 which is 0. Notice that our limit is constant, right? So it is a finite value and based on the test, whenever we get a limit of a finite value, we can conclude that the sequence converges. And we're going to report the limit, limit. As n approaches infinity of a n is equal to 0, that's our final answer, and thank you for watching.

Convergence and DivergenceWhich of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.aₙ = (n + 3) / (n² + 5n + 6)

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Convergence and Divergence
Which of the sequences {aₙ} in Exercises 31–100 converge, and which diverge? Find the limit of each convergent sequence.
aₙ = (n + 3) / (n² + 5n + 6)