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Chapter 7: Q. 57 (page 592)

In Exercises 55鈥� 58 use the ratio test in Theorem 7.6 to analyze the monotonicity of the given sequence.

3k2路4路6路路路(2k)

Short Answer

Expert verified

The given sequence is strictly decreasing fork鈮�1.

Step by step solution

01

Step 1. Given Information.

The given sequence is3k2路4路6路路路(2k).

02

Step 2. Use the ratio test. 

To analyze the monotonicity of the given sequence we will use the ratio test.

Let the general term of the sequence is ak=3k2路4路6路路路2k.

So, the term ak+1is

ak+1=3k+12路4路6路路路2k2k+1.

According to the ratio test,

localid="1649233002767" ak+1ak=3k+12路4路6路路路2k2k+13k2路4路6路路路2k=3k路33k2k+2=32k+2

Now, ak+1ak<1fork鈮�1.

Thus, the given sequence is strictly decreasing fork鈮�1.

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