Chapter 9.4, Problem 41E

Solution

To state: Whether the statement is true or false. If all terms of a series are positive, the series sum to a positive number.

The statement is false.

Given information:

It is given that if all terms of a series are positive, the series sum to a positive number.

Explanation:

Let’s consider a geometric series,

  $\frac{1}{48}+\frac{1}{16}+\frac{3}{16}+\frac{9}{16}+\cdots$

The common ratio ( r ) is:

  $\begin{array}{l}r=\frac{\frac{1}{48}}{\frac{1}{16}},r=3\\ r=\frac{\frac{3}{16}}{\frac{1}{16}},r=3\end{array}$

A geometric series ${\displaystyle {\sum }_{k=1}^{\infty }a\cdot r^{k-1}}$ , converges if and only if $\left|r\right|<1$ .If it does converge, the sum: $S=\frac{a_1}{1-r}$ .

For the given series the value of $a_1$ is $\frac{1}{48}$ and $r\text{is}3$ . As the common ratio $\left|r\right|>1$ , then the given series diverges. It means the sum of the series does not exist even if it consist all positive numbers.

Therefore, the given statement if all terms of a series are positive, the series sum to a positive number is false.