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 = (ln 3 - ln 2) + (ln 4 - ln 3) + (ln 5 - ln 4) +....+ (ln (n - 1) - ln (n - 2)) + (ln n - ln (n - 1))

Expert Solution

Step 1

The given sequence,

$a_{n} = (\ln 3 - \ln 2) + (\ln 4 - \ln 3) + (\ln 5 - \ln 4) + . . . . . + (\ln (n - 1) - \ln (n - 2)) + (\ln (n) - \ln (n - 1))$

We have to check whether the sequence is convergent or divergent.

Step 2

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

Now, the given sequence,

$a_{n} = (\ln 3 - \ln 2) + (\ln 4 - \ln 3) + (\ln 5 - \ln 4) + . . . . . + (\ln (n - 1) - \ln (n - 2)) + (\ln (n) - \ln (n - 1))$

Simplify the expression,

$\begin{array}{rcl} a_{n} & = & (\ln 3 - \ln 2) + (\ln 4 - \ln 3) + (\ln 5 - \ln 4) + . . . . . + (\ln (n - 1) - \ln (n - 2)) + (\ln (n) - \ln (n - 1)) \\ = & \ln 3 - \ln 2 + \ln 4 - \ln 3 + \ln 5 - \ln 4 + . . . . . + \ln (n - 1) - \ln (n - 2) + \ln (n) - \ln (n - 1) \\ = & \ln (n) - \ln 2 \\ = & \ln (\frac{n}{2}) \\ a_{n} & = & \ln (\frac{n}{2}) \end{array}$

Now, take the limit as $n \to \infty$ .

$\begin{array}{rcl} \lim_{n \to \infty} a_{n} & = & \lim_{n \to \infty} \ln (\frac{n}{2}) \\ = & \ln (\infty) \\ = & \infty \end{array}$

Here the limit $\lim_{n \to \infty} a_{n}$ is infinite.

Thus the sequence is divergent.

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