Answered: Which of the sequences {an} converge,…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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

Expert Solution

Step 1

The given sequence $a_{n} = \frac{n + 3}{n^{2} + 5 n + 6}$ .

We have to check the convergence of the sequence and find its limit if it is convergent.

Step 2

Sequence convergent:

If the sequence of real numbers $\{a_{n}\}$ is convergent, then the limit of the sequence $\lim_{n \to \infty} a_{n}$ exits finitely.

The given sequence $a_{n} = \frac{n + 3}{n^{2} + 5 n + 6}$ .

Now,

$\begin{array}{rcl} \lim_{n \to \infty} a_{n} & = & \lim_{n \to \infty} \frac{n + 3}{n^{2} + 5 n + 6} \\ = & \lim_{n \to \infty} \frac{n + 3}{n^{2} + 2 n + 3 n + 6} \\ = & \lim_{n \to \infty} \frac{n + 3}{n (n + 2) + 3 (n + 2)} \\ = & \lim_{n \to \infty} \frac{n + 3}{(n + 2) (n + 3)} \\ = & \lim_{n \to \infty} \frac{1}{n + 2} \\ = & \frac{1}{\infty} \\ \lim_{n \to \infty} a_{n} & = & 0 \end{array}$

Since the limit $\lim_{n \to \infty} a_{n} = 0$ exits finitely, the sequence $a_{n} = \frac{n + 3}{n^{2} + 5 n + 6}$ is convergent.

The limit of the sequence is $0$ .

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