Chapter 10.1, Problem 67E

To determine

Todetermine:The given statement is true or false. Justify your answer.

Answer to Problem 67E

The given statement is false.

Explanation of Solution

Given information:

Statement:The series $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots$ is diverges.

Methodused:

Theorem:Let ${\displaystyle \sum _{n=0}^{\infty }{c}_n}$ be a given series and L be a positive number such that $\underset{n\to \infty }{\lim }\frac{c_{n+1}}{c_n}=L$ , then if $L<1$ the series converges and if $L>1$ the series diverges.

The given series is ${\displaystyle \sum _{n=0}^{\infty }{c}_n}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots +\frac{1}{2^n}+\cdots$ .

  $\Rightarrow {\displaystyle \sum _{n=0}^{\infty }{c}_n}={\displaystyle \sum _{n=0}^{\infty }\frac{1}{2^n}}$

  $\Rightarrow c_n=\frac{1}{2^n}$

Thus, we have $L=\underset{n\to \infty }{\lim }\frac{c_{n+1}}{c_n}=\underset{n\to \infty }{\lim }\frac{\frac{1}{2^{n+1}}}{\frac{1}{2^n}}$

  $\Rightarrow L=\frac{2^n}{2^{n+1}}=\frac{1}{2}<1$

Thus, the series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}=1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\cdots +\frac{1}{2^n}+\cdots$ converges.

Hence, the given statement is false.

Conclusion:

The given statement is false.