Chapter 12.4, Problem 8CFU

To determine

To determine whether given series is convergent or divergent.

Answer to Problem 8CFU

By comparison test, the series is divergent.

Explanation of Solution

Given information:

  $\frac{1}{4}+\frac{5}{16}+\frac{3}{8}+\frac{7}{16}+\mathrm{...}$

Concept used:

Comparison Test: A series of positive terms is convergent if, for $n>1$ , each term of the series is equal to or less than the value of the corresponding term of some convergent series of positive terms.

Similarly, a series of positive terms is divergent if, for $n>1$ , each term of the series is equal to or greater than the value of the corresponding term of some divergent series of positive terms.

Calculation:

  $\frac{1}{4}+\frac{5}{16}+\frac{3}{8}+\frac{7}{16}+\mathrm{...}$

Make the denominator same of all the terms.

  $\frac{4}{16}+\frac{5}{16}+\frac{6}{16}+\frac{7}{16}+\mathrm{...}$

Find the general term $a_n$ .

  $a_n=\frac{n+3}{16}$

Consider the divergent series $\frac{1}{16}+\frac{2}{16}+\frac{3}{16}+\mathrm{...}$

The general term of this series is $\frac{n}{16}$ .

For all $n$ ,

  $\frac{n+3}{16}>\frac{n}{16}$

Hence, the series $\frac{1}{4}+\frac{5}{16}+\frac{3}{8}+\frac{7}{16}+\mathrm{...}$ is divergent.