Q. 7 Limits of sequences: Determine w... [FREE SOLUTION]

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Chapter 7: Q. 7 (page 656)

Limits of sequences: Determine whether the sequences that follow are bounded, monotonic and/or eventually monotonic.
Determine whether each sequence converges or diverges. If the sequence converges, find its limit.
$\{\frac{k!}{1 路 3 路 5 路路路 (2 k - 1)}\}$

Short Answer

Expert verified

The sequence is montonically decresing and bounded.
The limit of the sequence is 0.

Step by step solution

Step 1. Given Information

The given sequence is $\{\frac{k!}{1 路 3 路 5 路路路 (2 k - 1)}\}$ .

Step 2. Apply Ratio Test

Find the ratio $\frac{a_{k + 1}}{a_{k}}$ .

$\begin{array}{rcl} \frac{\frac{(k + 1)!}{1 路 3 路 5 路路路 (2 k - 1) (2 k + 1)}}{\frac{k!}{1 路 3 路 5 路路路 (2 k - 1)}} & = & \frac{(k + 1)!}{1 路 3 路 5 路路路 (2 k - 1) (2 k + 1)} 脳 \frac{1 路 3 路 5 路路路 (2 k - 1)}{k!} \\ = & \frac{k + 1}{2 k + 1} \end{array}$

As $\frac{k + 1}{2 k + 1} < 1$ for all k, the sequence is strictly monotonically decreasing.

Step 3. Check for Boundedness and Limit

All the terms of the sequence are positive and the sequence is decreasing, it is bounded below by zero, and because the sequence is strictly decreasing, the first term, 1 will be the upper bound for the sequence.
Therefore, the sequence is bounded.
The limit of the sequence will be 0, that is the lower bound of the decreasing sequence,

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91影视

Limits of sequences: Determine whether the sequences that follow are bounded, monotonic and/or eventually monotonic.
Determine whether each sequence converges or diverges. If the sequence converges, find its limit.
$\{\frac{k!}{1 路 3 路 5 路路路 (2 k - 1)}\}$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Apply Ratio Test

Step 3. Check for Boundedness and Limit

One App. One Place for Learning.

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91影视

Limits of sequences: Determine whether the sequences that follow are bounded, monotonic and/or eventually monotonic.Determine whether each sequence converges or diverges. If the sequence converges, find its limit.k!1路3路5路路路(2k-1)

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Apply Ratio Test

Step 3. Check for Boundedness and Limit

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Geometry

Theoretical and Mathematical Physics

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Limits of sequences: Determine whether the sequences that follow are bounded, monotonic and/or eventually monotonic.
Determine whether each sequence converges or diverges. If the sequence converges, find its limit.
$\{\frac{k!}{1 路 3 路 5 路路路 (2 k - 1)}\}$